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ELEMENTS 



Differential a 




ral 



CALCULUS, 



By a New Method, Founded on the True System of 

Sir Isaac Newton, without the Use of 

Infinitesimals or Limits. 



A J 



By C? P. BUCKINGHAM, 

AUTHOR OF THE PRINCIPLES OF ARITHMETIC ; FORMERLY ASSISTANT PROFESSOR OF 

NATURAL PHILOSOPHY IN THE U. S. MILITARY ACADEMY, AND PROFESSOR OF 

MATHEMATICS AND NATURAL PHILOSOPHY IN KENYON COLLEGE, OHIO. 



CHICAGO: 

S. C. GRIGGS & COMPANY. 

1875- 




Entered according to act of Congress, in the year 1875, by 

S. C. GRIGGS & CO., 

in the office of the Librarian of Congress at Washington, District of Columbia, 



r LAK ES IOC 

/pot 



Preface. 



" The student of mathematics, on passing from the lower 
branches of the science to the infinitesimal analysis, rinds 
himself in a strange and almost wholly foreign department 
of thought. He has not risen, by easy and gradual steps, 
from a lower to a higher, purer, and more beautiful region of 
scientific truth. On the contrary, he is painfully impressed 
with the conviction, that the continuity of the science has 
been broken, and its unity destroyed, by the influx of prin- 
ciples which are as unintelligible as they are novel. He 
finds himself surrounded by enigmas and obscurities, which 
only serve to perplex his understanding and darken his aspi- 
rations after knowledge."* 

He finds himself required to ignore the principles and 
axioms that have hitherto guided his studies and sustained 
his convictions, and to receive in their stead a set of notions 
that are utterly repugnant to all his preconceived ideas of 
truth. When he is told that one quantity may be added to 
another without increasing it, or subtracted from another 
without diminishing it — that one quantity may be infinitely 
small, and another infinitely smaller, and another infinitely 

* Bledsoe — Philosophy of Mathematics. 



IV PREFACE. 

smaller still, and so on ad infinitum — that a quantity may 
be so small that it cannot be divided, and yet may contain 
another an indefinite, and even an infinite, number of times — 
that zero is not always nothing, but may not only be some- 
thing or nothing as occasion may require, but may be both at 
the same time in the same equation — that two curves may 
intersect each other and yet both be tangent to a third curve 
at their point of intersection,* it is not surprising that he 
should become bewildered and disheartened. Nevertheless, 
if he study the text books that are considered orthodox in 
this country and in Europe he will find some of these 
notions set forth in them all ; not, indeed, in their naked 
deformity, as they are here stated, but softened and made as 
palatable as possible by associating them with, or concealing 
them beneath, propositions that are undoubtedly true. 

It is, indeed, strange that a science so exact in its results, 
should have its principles interwoven with so much that is 
false and absurd in theory ; especially as all these absurdi- 
ties have been so often exposed, and charged against the 
claims of the calculus as a true science. It can be accounted 
for only by the influence of the great names that first adopted 
them, and the indisposition of mathematicians to depart 
from the simple ideas of the ancients in reference to the 
attributes of quantity. They regard it as merely inert, 
either fixed in value or subject only to such changes as may 
be arbitrarily imposed upon it. But when they attempt to 
carry this conception into the operations of the calculus, 
and to account for the results by some theory consistent with 

* As in the case of envelopes. 



PREFACE. V 

this idea of quantity they are inevitably entangled in some 
of the absurd notions that have been mentioned. Many 
efforts have indeed been made to escape such glaring incon- 
sistencies, but they have only resulted in a partial success 
in concealing them. , 

To clear the way for a logical and rational consideration 
of the subject, we must begin with the fundamental idea of 
the conditions under which quantity may exist. We must, 
for the purposes of the calculus, consider it not only as ca- 
pable of being increased or diminished ; but also as being 
actually in a state of change. It must (so to speak) be vital- 
ized^ so that it shall be endowed with tendencies to change its 
value ; and the rate and direction of these tendencies will 
be found to constitute the ground work of the whole system. 
The differential calculus is the science of rates, and its 
peculiar subject is quantity in a state of change. 

It is an error, therefore, to suppose, as has often been 
said, that the " 7'eductio ad absurdum" or " method of ex- 
haustion/' of the ancient mathematicians, contained the 
germ of the differential calculus. This hallucination has 
arisen from the same source as the false notions before 
mentioned. That peculiar attribute of quantity upon which 
the transcendental analysis was built, never found a place 
among the ideas of the Greek Philosophers ; and even Leib- 
nitz, the competitor of Newton for the honor of the inven- 
tion, and who was the first to construct a system of rules 
for the analytical machinery of the science, never got beyond 
the ancient conception of the conditions of quantity, and, 
therefore, gave, says Comte, " an explanation entirely erro- 



VI PREFACE. 

neous " — he never comprehended the true philosophical 
basis of his own system. 

The only original birth-place of the fundamental idea of 
quantity which forms the true germ of the calculus, was in 
the mind of the immortal Newton. Starting with this idea, 
the results of the calculus follow logically and directly 
through the beaten track of mathematical thought, with that 
clearness of evidence which has ever been the boast of 
mathematics, and which leaves neither doubt nor distrust 
in the mind of the student. 

To develop this idea is the object of this work. 

C. P. BUCKINGHAM. 
Chicago, Jan. i, 1875. 



Contents. 



INTRODUCTION. 

PAGE. 

Objects of Mathematical Study among the Ancient Philosophers II 

Method of Descartes _ 12 

The Differential Calculus — _ 14 

Two General Methods 16 

Infinitesimal Method _ .„ 18 

Results are True while the Method is False 20 

Method of Limits ._ 22 

Newton's Defense of his Lemma 23 

Opinions of Comte, Lagrange and Berkeley 24 

Fundamental Error of Both Systems _ __ 31 

The True Method of Newton 33 

The Foundation of that Method . „.„ .__ 34 

Explanatory Letter of the Author .. ......oo* 38 



PART I. 



DIFFERENTIAL CALCULUS. 

Section I. Definitions and First Principles „ 49 

Variables Defined. --__- 49 

Rate of Variation. __ „ __ 50 

Differentials 51 

Constants . 52 

Functions 53 

Section II. Differentiation of Functions . 56 

Sign of the Differential 56 

Differential of a Function Consisting of Terms 58 

Forms of Algebraic Terms 60 

Differential of a Product of Two Variables 63 

Geometrical Illustration 65 

Pifferentials of Fractions „ 6§ 



Vlll CONTENTS. 

PAGE. 

Differential of the Tower of a Variable 71 

" Root " " _ 73 

" " Function of Another Function __ 74 

Section III. Successive Differentials _ 

Maclaurin's Theorem 85 

Taylor's Theorem 89 

Identity of Principle in Both Theorems (note) 93 

Section IV. Maxima and Minima 95 

Method of Finding by Substitution 96 

Meaning of a Maximum 97 

Method of Finding by the Second Differential 98 

Use of Other Differential Coefficients 99 

Examples Illustrating Different Cases 100 

Analytical Demonstration of the General Rule _ 105 

Section V. Application of the Calculus to the Theory of Curves. 125 

Definition of a Line _ 125 

Why the Line Becomes a Curve 126 

Direction of the Tangent Line ._ 127 

Real Meaning of the Differential Equation 129 

Sign of the First Differential Coefficient 132 

Section VI. Differentials of Transcendental Functions. 135 

Differential of av ___ 135 

" of the Logarithm of a Variable __ 137 

" " Sine of an Arc 139 

" " Cosine of an Arc 140 

" " Tangent of an Arc 141 

" " Secant of an Arc _ 142 

" " Versed Sine of an Arc 143 

" " Arc ___ 143 

Signification of the Differentials of Circular Functions 144 

Values of Trigonometrical Lines 148 

Section VII. Tangent and Normal Lines to Algebraic Curves 150 

Length of Subtangent to any Curve 152 

" Tangent " " 153 

" Subnormal " " 153 

" Normal " " 154 

Application of the formulas 154-156 

SECTION VIII. Differentials of Curves 157 

Differential Plane Surfaces Bounded by Curves. . _ 158 

" Surfaces of Revolution _ 162 



CONTENTS IX 

PAGE. 

Differential Solids of Revolution 165 

Section IX. Polar Curves 168 

Tangents to Polar Curves __ 168 

Differential of the Arc of a Polar Curve 171 

Subtangent to a Polar Curve 171 

Tangent " " " 171 

Subnormal " " " 172 

Normal " u " 172 

Surface Bounded by a Polar Curve 173 

Spirals __ 173 

Spiral of Archimedes __ 174 

Hyperbolic Spiral _- 176 

Logarithmic " — 179 

Section X. Asymptotes ___ 182 

How to Find Them__. _._ 183 

Examples 183-186 

Section XI. Signification of the Second Differential 187 

Sign of the Second Differential Coefficient 188 

Value " " " " _ _ 190 

Section XII. Curvature of Lines _ __ 192 

Measure of Curvature _ ^ 192 

4 Contact of Curves , 193 

Constants in the Equation of a Curve __ 195 

Osculatrix to a Curve 199 

Radius of Curvature __ _ _ 202 

Section XIII.. Evolutes___ 207 

Properties of the Evolute 207 

To Find the Equation of the Evolute 210 

Section XIV. Envelopes _ 216 

Definition of an Envelope ___ 218 

How to Obtain its Equation 218 

Section XV. Application of the Calculus to the Discussion of Curves 229 

The Cycloid _ 229 

Properties of the Cycloid 230 

Logarithmic Curve 238 

Section XVI. Singular Points _ 243 

Maxima and Minima 243-248 

Cusps _ 248 

Conjugate Points 253 

Multiple Points „ ....,- 254 



CONTENTS. 

PART II. 



INTEGRAL CALCULUS. 

PAGE. 

Section I. Principles of Integration. _ 261 

Integration of Compound Differential Functions 263 

11 Monomial " " 263 

" Particular Binomial Differentials 266 

" Rational Fractions . 267 

" Between Limits 277 

" by Series 278 

" of Differentials of Circular Arcs 279 

Section II. Integration of Binomial Differentials 282 

Integration of Particular Forms 283-288 

" by Parts 291 

Formulas for Reducing Exponents 293-303 

Section III. Application of the Calculus to the Measurement of 

Geometrical Magnitudes 304 

Rectification of Curves _ __ 305 

Quadrature of Curves 312 

Surfaces of Revolution _ ' 325 

Cubature of Solids , 329 



APPENDIX. 



GEOMETRICAL FLUXIONS, 

Principles of the Calculus Applied Directly to Geometrical Magnitudes 337 

To Find the Area of a Circle 338 

" " " Convex Surface of a Cone 339 

" a " Volume of a Cone . 340 

" " " Area of the Surface of a Sphere 341 

" " " Volume of a Sphere 342 



Introduction. 



THE PHILOSOPHY OF THE CALCULUS. 

Among the ancient philosophers, the objects of mathemat- 
ical study were confined exclusively to the solution of deter- 
minate problems — that is, every quantity concerned had a 
determinate value, either known or unknown. Devoting 
* themselves principally to Geometry, they sought to deter- 
mine the exact measurement of lines, surfaces, solids and 
angles, in terms of fixed and known quantities. 

The later methods of the algebraists did not change the 
ultimate object of their researches. All their problems 
were still determinate. Their conditions were definite, and 
the result certain. This, which may properly be character- 
ized as the static phase of mathematics, continued for two 
thousand years to guide, control and circumscribe the labors 
of the mathematical student. There was but little advance 
in the discovery of mathematical truth ; none had the bold- 
ness to strike out a new method of investigation, or apply 
themselves to the solution of any but determinate problems. 
Algebra had indeed been successfully applied to geometry, 
but it was only the analytical method of stating arguments 
that had been used in ordinary language for centuries* 



Xll INTRODUCTION. 

Such was the condition of the science up to the time 
when the brilliant genius of Descartes seized upon a new 
idea, and boldly followed its lead until he developed a sys- 
tem whose results have astonished and delighted the world. 

Breaking away from the idea of determinate values and 
absolute conditions, he adopted that of dependent condi- 
tions and relative values, which no longer fixed unchangea- 
bly the quantities sought, but gave them a wide range, so 
that within certain limits they could have all possible values. 
Hence they were called variables, while those quantities 
whose values were fixed were called constants. 

In every equation containing a single unknown quantity 
the value of that quantity is absolutely fixed by the condi- 
tions expressed in the equation. If we have two unknown 
quantities, and two equations, or sets of conditions, both 
values are still fixed. If the higher power of the unknown 
quantity is involved, the number of values is greater, but 
they are equally fixed and certain. This idea of fixedness 
of value underlies all algebraic operations of an ordinary 
kind. 

Now suppose we have two unknown quantities in one equa- 
tion, with no other conditions given than those expressed by 
the equation itself. In that case the values of both quanti- 
ties are absolutely indeterminate. But if we know or 
assume any specific value for one we can at once determine 
the corresponding value of the other; so that while the 
equation will give the independent value of neither quan- 
tity, it will give the simultaneous values of both ; and these 
values will have a certain range or locus, which is in fact the 
true solution of the equation — the path, so to speak, through 
which the simultaneous values range. 

In some equations the range of values is limited for both 
variables, so that if a value be assigned to one beyond the 
limit, that of the other becomes imaginary ; in other cases 



INTRODUCTION. Xlll 

the value of one only is limited, while in others again the 
values of both are absolutely unlimited; any value of one 
giving a corresponding real value for the other. 

Since the values of these variables are thus dependent on 
each other, the equation expressing this dependence may be 
considered as containing the law of their mutual relations, 
and the fundamental ideas of Descartes was to exhibit in his 
equation the conditions or law which confined the two vari- 
ables to their prescribed range of values. This idea was 
something new, distinct and well defined, and a clear ad- 
vance beyond the methods of the ancients. 

But the labors of Descartes would have been of little 
value had he proceeded no farther than we have indicated. 
In fact this was but a part of his invention, of which the 
specific object was a method of investigating questions 
of Geometry. To complete this purpose, he devised a new 
and beautiful method of representing magnitudes, to which 
his algebraic equations could be applied. In algebra, all 
4 values are estimated by their remoteness from zero. In 
order to make a. general application of algebraic symbols to 
geometry, it was necessary that the value of every line rep- 
resented by his variables should be estimated from a com- 
mon origin corresponding with zero ; and as every point in 
a plane surface requires two values to fix its position, two 
such origins became necessary to his system, in order to 
represent plain figures ; and these were found in two right 
lines, lying in the plane of the object to be represented, and 
intersecting in a known angle — generally a right angle, 
From these two lines all values or distances to points were 
estimated ; the positive on one side and the negative on the 
other of each line ; while for points in the lines, one of the 
values would of course be zero. 

Having then a method of representing the position of a 
point by algebraic symbols, it was easy to apply his analysis 



XIV INTRODUCTION. 

to the representation of lines, by making the locus or range 
of simultaneous values of the variables to correspond with 
the locus of the points in the line — that is, with the line 
itself 

Thus the method of Descartes was two-fold — the alge- 
braic idea of two variables in one equation with a range of 
simultaneous values, and the geometrical idea of coordinate 
representation, and these two being adapted to each other, 
united to form a method of investigating, in an easy and sim- 
ple manner, questions of geometry which had taxed the 
utmost powers of the ancients. 

Upon the foundation thus laid by Descartes has arisen 
the Differential Calculus. Not that the calculus in its purely 
abstract conception is especially related to geometry. On 
the contrary its analysis is adapted to investigations in all 
those questions where the quantities are variable and the 
conditions can be analytically expressed. But it was in con- 
nection, with problems of geometry that its methods were first 
discovered, and for a long time applied through the Cartesian 
system ; and even now geometry is the principal arena upon 
which the triumphs of the calculus are displayed. 

The invention itself has many peculiarities both in its 
history and substance. It was not a result produced by 
means of ordinary scientific investigation — by a discovery 
of fundamental principles and a careful elaboration of those 
principles until they grew into a perfect science. On the 
contrary these results appeared rather as remarkable phe- 
nomena, discovered more by accident than by logical deduc- 
tion. Newton seems indeed to have had an indistinct per- 
ception of the principles lying at the foundation, but he has 
nowhere given a clear and satisfactory account of them ; while 
the explanation given by Leibnitz proved his utter ignorance 
of the true theory of his own system. 

Thus was a most important branch of mathematics invented 



INTRODUCTION. XV 

almost simultaneously by two of the most distinguished men 
of the age, without any clear and fundamental principle for 
its foundation. Its operations were accepted as undoubtedly 
reliable, not because its principles were sound, but because 
its results were undeniably true. This could not be disputed, 
and hence mathematicians were not so eager, at first, to 
establish a logical basis for the new science, as to extend its 
operations into new fields of discovery. Attempts were, at 
length, made to assign a rational principle which would 
account for these extraordinary results, but although each 
theory has had for its advocates many of the most distin- 
guished mathematicians, yet each one has had as many 
and as distinguished opponents. No one has secured the 
universal approval of the scientific world, and, therefore 
no one was founded in mathematical truth ; for no pro- 
position is worthy of the name that does not command 
the unqualified assent of every mind by which it is fully 
comprehended. 

The conceptions of the calculus were so subtle, its pro- 
cesses so mysterious, and its results so astounding that 
mathematicians began to look upon its ideas as not subject 
to the ordinary laws of thought and the rigid rules of 
logic. The inconsistencies and absurdities which were 
often propounded were regarded as only mysterious and 
incomprehensible j when quantities refused to obey the laws 
that had always hitherto controlled them, they were called 
infinitesimals, and thus released from all subjection to 
establish axioms. Of course theories were not wanting, but 
they did little more than give variety to what was, after all, 
the same compound of false premises and illogical conclu- 
sions. We are told by M. Comte that the science is as yet 
in a " provisional state," and that it is necessary to study all 
the principal methods in order to have even an approximate 
understanding of it. 



XVI INTRODUCTION. 

I shall, however, confine myself to the consideration of 
the two principal conceptions attributed to its inventors, 
and to some more recent modifications of them. 

The advocates of these two methods have approached the 
subject from the same direction, but the theories involved in 
their demonstrations are fundamentally different. These I 
propose to examine, and to show that as theories they are 
fatally defective ; that a fundamental error underlies every 
form in which they have been proposed, and must vitiate 
any theory based upon the leading idea through which they 
approach the subject. 

By the invention of Descartes very many geometrical 
problems were beautifully solved, but there were some for 
which equations between the direct functions of the varia- 
bles would not suffice ; such as the length of a curve, the 
amount of its curvature, and others of a similar kind. To 
form equations in which such values could be introduced, 
it became necessary to represent the variables indirectly, 
using, instead of the actual value, a function of that value. 
This function is what is called the differential of the variable, 
and the true philosophical relation which it sustains to its 
actual value has been the subject of controversy from the 
beginning. The particular application of the calculus, which 
will most clearly and exactly illustrate the various systems, 
is to determine from the equation of the curve the direction of its 
tangent at any point. This process involves the fundamental 
principles of the science, and an examination of it will 
afford the best means of investigating the different theories 
that have been advanced to account for the exact truth of 
the results obtained. 

Let APC (Fig. A) be a curve, and let AM and AN be the 
coordinate axes to which it is referred. Suppose the line SD 
to be drawn tangent to the curve at the point P. The 
problem is to find from the equation of the curve an ex- 



INTRODUCTION. 



XV. 1 



pression for the value of the 
angle PSB. Now the tangent 
of this angle is 

PB 

SB 
S being the point of, intersec- 
tion of the tangent line with the 
axis of abscissas, and B the ] 
foot of the ordinate through 
the point of contact. Let BB' be an increment added to the 
abscissa, and B'P' the ordinate corresponding to the abscissa 
thus increased. Draw PE parallel to AM, intersecting B'P' 
in E ; then P'E is the corresponding increment of the ordi- 
nate BP. Draw the cord P'P, and let D be the point where 




B 

Fig. A. 



B' M 



the tangent intersects the ordinate B'P'. 

PB DE 



Since 



SB PE 
the problem is reduced to finding the ratio of PE to DE. 
Now we can easily find from the equation of the curve, the 
line P'E corresponding to any increment BB' or PE of the 
abscissa AB But DE is the line we need, and to pass from 
P'E to it is the specific part of the operation which involves 
the fundamental principles of the Calculus. The two prin- 
cipal methods of doing this we will now examine. 

THE INFINITESIMAL METHOD. 



The fundamental propositions or principles of this method 
are : First. " That we may take indifferently the one for the 
other, two quantities which differ from each other by an infi- 
nitely small quantity, or (what is the same thing) that a 
quantity which is increased or diminished by another infinity 
less than itself can be considered as remaining the same. 
Second. That a curved line may be considered as an assem- 
2 



XV1U INTRODUCTION. 

blage of an infinity of right lines each infinitely small, or 
(what is the same thing) as a polygon with an infinite num- 
ber of sides infinitely small, which determine, by the angle 
which they make with each other, the curvature of the 
line." 

In order to understand clearly how these propositions 
apply to the solution of the problem we will consider it 
analytically. Suppose the equation of the curve to be 

y=x 2 (i) 

If we take AB — x we have BP=yy and if we add to x an 
increment, BB' (which we will call /i), the corresponding 
ordinate will be P'B', which we will designate by j/, and 
P'E will be equal to y — y. Having added h to x, the new 
state of the equation will be 

y =(x+/i) 2 =x 2 +2/1X+/1 2 (2) 

Subtracting (1) from (2) we have 

y — y == 2/ix -\-7i 2 
or dividing by h 

y-y _ , 7 p'e 

h 2X ~t /1 pe 
Now as h diminishes in value, this ratio approaches the 
value of 2x ; and if h is made infinitely small, it may by 
the first proposition above stated, be set aside as not affect- 
ing the value of the second member of the equation, which 
then becomes 2X. In the meantime the curve PP' will have 
become infinitely small, and therefore by the second propo- 
sition may be considered a straight line, coincident with the 
tangent, that is with PD, so that P' E has become the same 
as DE, and the equation 

P'E 

r^- = 2X+/l 

has become 

DE 



PP 2X 



TNTRODUCTTOV. XIX 

and thus the angle made by the tangent line with the axis 
of abscissas is determined. Such is the infinitesimal 
theory. 

The propositions upon which this theory is founded can- 
not be admitted as true, neither is the demonstration con- 
clusive. To whatever extent h may be reduced, even to an 
infinitesimal (if it is possible to conceive such a thing), the 
two expressions 2x and 2x'-\-/i can never be equal so long as 
h is anything. We must either admit this or abandon the use 
of our reason. If h becomes nothing in one number of the 
equation, it must do so in the other, that is, PE=/^ must 
also become zero, and so must P'E, and we have instead of 

DE . o 

-^pr z=z 2X simply ~—2X 
PE L J o 

an expression which certainly amounts to nothing, unless we 

can show that the tangent of the angle DPE or PSB is 

o 
equal to —. 

Again it is impossible that a curve should ever be consid- 
ered as a polygon. The very definition of a curve is sim- 
ply that which distinguishes it from a straight line. Had 
the ancient geometers been willing to admit this principle, 
how easily could they have avoided the tedious and labori- 
ous " reductio ad absurdum." But they were too conscien- 
tious and exact to admit the possibility of establishing truth 
by even a doubtful principle. " In more modern times the 
greatest mathematicians and philosophers have, indeed, 
emphatically condemned the notion, that a curve is or can % 
be made up of right lines, however small. Berkeley, the 
celebrated Bishop of Cloyne, and his great antagonist, Mac- 
laurin, both unite in- rejecting this notion as false and unten- 
able. Carnot, D'Alembert, Legrange, Cauchy, and a host 
of other illustrious mathematicians, deny that the circum- 
ference of a circle, or any other curve, can be identical with 



XX INTRODUCTION. 

the periphery of any polygon whatever." So repugnant is 
this proposition to the fixed and fundamental conceptions of 
geometry, that it has been doubted and denied in all ages 
by the most competent thinkers and judges. 

But notwithstanding all this, what shall we say when we 
find that the equation 

DE 

PE ~ 2X 
is not merely an approximation to the truth, but that it is 
perfectly and exactly true. We have said that errors have 
manifestly been committed in arriving at this result. The 
advocates of the system point to the result and say behold 
the proof that we are right. The explanation of this seem- 
ing mystery was made long ago by Bishop Berkeley. " For- 
asmuch," says he, " as it may perhaps seem an unaccount- 
able paradox that mathematicians should deduce true prop- 
ositions from false principles, be right in the conclusion, and 
yet err in the premises, I shall endeavor particularly to 
explain why this may come to pass, and show how error may 
bring forth truth, though it cannot bring forth science." He 
then proceeds to give an illustration, for which we will sub- 
stitute a similar one adapted to the figure we have chosen. 

It will be perceived that the curve we have taken is a par- 
abola whose axis is that of ordinates, and whose parameter 
is i. Now if we consider the curve PP' as infinitely small 
and a straight line, the angle P'PE would be the angle we 
are seeking ; but since the curve can never become a straight 
line, the increment P'E must always be too great, and the 
point P' must always be above the tangent line; so that the 
error arising from taking PP' as a part of the tangent line 
will be equal to P'D, or that part of tke increment of the 
ordinate which lies between the tangent and the curve. But 
we have found 

y —y z=L 2xk -\-h % = P'E 



INTRODUCTION. XXI 



and since from the nature of the parabola 



x 

2 



we have from similar triangles 

DE : PE : : PB : SB 



or 



DE : h : :y : — : : x~ 



2 



DE=2^ 



hence 

and 

P'D^P'E— DE = 2^+/i 2 -2^=A 8 
Here then we find that by throwing out ft 2, we exactly cor- 
rect the error arising from taking the curve PP r as a straight 
line ; and reduce the line P'E=j/ — y to DE the line we are 

DE 

seeking to fix the value of ~Sp~> which is the tangent of the 

angle which the tangent line makes with the axis of abscissas. 
Thus the infinitesimal method arrives at the true result, 
not because its principles are true, nor because its errors are 
small, but because they are, whether great or small, exactly 
equal, and exactly cancel and destroy each other. This theory 
then is not only false but unnecessary. The false proposi- 
tions with which it sets out are as unnecessary as they are 
absurd. The errors may as well be great as small. The 
system is but a mere artifice, which " by means of signs and 
symbols and false hypotheses, has been transformed into the 
sublime mystery of the transcendental analysis." We dis- 
miss it, therefore, with the remark, that to admit and accept 
an error, even infinitesimal, in mathematics, strips the sci- 
ence of its chief glory, and introduces darkness and doubt 
where only the pure light of truth should prevail — in fact 
it opens the door for anarchy in all science, and unsettles 
the very foundations of all knowledge. 



XX11 INTRODUCTION. 



THE METHOD OF LIMITS. 



The philosophical principle on which this method is 
based, is thus stated by Sir Isaac Newton, in the enunciation 
of the first lemma in the first book of the Principia. 

" Quantities and ratios of quantities, which in any finite time 
converge continually to equality, and before the end of that time 
approach ?iearer the one to the other than by any given differ- 
ence, become ultimately equal." 

The principle here stated would be applied to the solution 

of our problem in the following manner. 

T , . DE . . 

1 ne ratio iS considered as the ultimate value of the ratio 

P'E ,, . 

p^ which approaches it as BB decreases, and coincides 

with it when B' has reached the point B, for then the points 

r 

y — y 

P' and D will have come together. Again — 7 — = 2x+h ap- 
proaches the value of 2x as h, or B'B, diminishes, and 
reaches that value when h becomes zero. 

P'E y-y 
Now since -pp-= — i — , and since by the lemma of New- 

P'E . DE 

ton pp becomes ultimately equal to "pp~, and since by the 

y ~y 

same lemma — i — becomes ultimately equal to 2x, at the 

DE 

same time and for the same reason, it follows that ^^ is 

' PE 

equal to 2X. 

It will here be seen that the point on which this demon- 

P'E 

stration turns is, that since the ratio ~^E r approaches the 

. DE 

ratio pp , as BB decreases, nearer than by any given dif- 



INTRODUCTION. XXlll 

ference, they become, according to the lemma, ultimately 

P'E y'-y DE 

equal; and the equation -5^"— — 1 — becomes p^ =2X 

which is its limit. 

The objection that lies immediately under the surface of 
this demonstration is, that when BB' or h has become zero, 

and we have for a result 

o . DE 

— = 2X instead of ~^r = 2X 

o BE 

Thus while we arrive at the desired value of our ratio, the 
ratio itself has lost all meaning, or at least all the attributes 
of quantity. 

The mind of Newton was too acute not to perceive the 
apparent absurdity involved in this application of his prin- 
ple, and he therefore gives the following explanation and 
defense of it : 

11 Perhaps it may be objected that there is no ultimate pro- 
portion of evanescent quantities ; because the proportion 
before the quantities have vanished is not ultimate, and 
when they have vanished is none. But by the same argu- 
ment it may be alleged, that a body arriving at a certain 
place, and there stopping, has no ultimate velocity ; because 
the velocity before the body comes to the place is not its 
ultimate, velocity; when it has arrived is none. But the 
answer is easy; for by the ultimate velocity is meant that 
with which the body is moved, neither before it arrives at its 
last place, and the motion ceases, nor after, but at the very 
instant it arrives : that is, the velocity with which it arrives 
at its last place, and with which the motion ceases. And in 
like manner, by the ultimate ratio of evanescent quantities, 
is to be understood the ratio of the quantities, not before 
they vanish nor afterwards, but with which they vanish." 

The illustration here given by Newton unfortunately 
throws no additional light upon the subject. It is certainly 



XXIV ■ INTRODUCTION - . 

no easier to conceive a body coming to a stop with any 
velocity, than it is to conceive of quantities vanishing with 
a ratio. Sir Isaac Newton is quite right when he says that 
the same argument may be alleged against both propositions, 
for both involve notions that are equally repugnant to our 
reason and consciousness ; one that there may be a ratio 
without quantities, and the other that there may be velocity 
without motion. In fact the illustration is, if anything, the 
more objectionable notion of the two, for the very reason 
why a body stops in its motion is because its velocity has 
expired and is gone. 

The argument based on Newton's lemma has been, by no 
means, universally received even among those mathemati- 
cians who reject the philosophy of Leibnitz. 

M. Comte, who views the method of limits with consider- 
able favor, says of it : " It is very far from offering such 
powerful resources for the solution of problems as the infin- 
itesimal method. The obligation it imposes of never con- 
sidering the increments of magnitudes separately and by 
themselves, nor even by their ratios, retards considerably the 
operations of the mind in the formation of auxiliary equa- 
tions. We may even say that it greatly embarrasses the 
purely analytical transformations/' 

Again in speaking of the course adopted by certain geom- 

eters, he says : " In designating by -7- that which logically 

Ay ' 
ought, m the theory of limits, to be denoted by L~ — n (that 

_. . Increment of v\ , . .. " , . 

is, Limit -7 ■ r 1, and in extending to all the other 

Increment of x' 

analytical conceptions this displacement of signs, they in- 
tended, undoubtedly, to combine the special advantages of 
the two methods, but, in reality, they have only succeeded 
in causing a vicious confusion between them, a familiarity 



INTRODUCTION. XXV 

with which hinders the formation of clear and exact ideas 
of either." 

Says Lagrange : " That method has the great inconven- 
ience of considering quantities in the state in which they 
cease, so to speak, to be quantities ; for although we can 
always well conceive the ratio of two quantities, as long as 
they remain finite, that ratio offers to the mind no clear and 
precise idea, as soon as its terms have become the one and 
the other nothing at the same time." 

But the objection to the lemma in question as a funda- 
mental principle of the calculus lies deeper than in its weak- 
ness and inefficiency „ The proposition carried to its legiti- 
mate results, overthrows the very system it is supposed to 
establish. Says the acute and candid Bishop Berkeley, in 
reply to Jurin : " For a fluxionist writing about mom entums, 
to argue that quantities must be equal because they have no 
assignable difference, seems the most injudicious step that 
could be taken ; it is directly demolishing the very doctrine 
you would defend. For it will thence follow that all hom- 
ogenous momentums are equal, and consequently the veloc- 
ities, mutations or fluxions proportional to these are likewise 
equal. There is, therefore, only one proportion of equality 
throughout which at once overthrows the whole system you 
undertake to defend." 

This is conclusive. If quantities that during any finite 
time constantly approach each other, and before the end of 
that time approach nearer than any given difference are 

P r E DE 

ultimately equal, then not only p^ and ~wft become ulti- 
mately equal, but PP', PD, PE, P'E and DE all become 
ultimately equal, for they all fulfill the conditions required by 

DE o 

lemma. Hence instead of p^ =2x, or even —=2x y we 

have as the logical sequence of this proposition 2x=i. ■ 



XXVI INTRODUCTION. 

Although the method of limits has generally been attrib- 
uted to Sir Isaac Newton, who was the author of the prop- 
osition which has served for its foundation, it is certain that 
this method as applied to the Differential Calculus, or method 
of fluxions, was not his. He has laid for that a very differ- 
ent foundation, as we shall see in due time, which has cer- 
tainly this merit, that there is nothing false nor absurd 
about it ; and if it does not clearly unravel the mysteries of 
the calculus, it places in cur hands the only clue by which 
we can do it for ourselves. 

METHOD OF LIMITS APPLIED TO THE DOCTRINE OF RATES. 

Prof. Loomis, of New Haven, has undoubtedly adopted 
the true conception of the nature of a differential. But he 
has unfortunately attempted to combine it with the method 
of limits, and has, therefore, become entangled in the same 
inconsistencies that we have already found to be inseparably 
attached to that method. 

It will be observed that in the following demonstration 
taken from his calculus, the question hinges on the ratio of 
the differential or rate of change of a variable to that of its 
square. And hence the demonstration although not arising 
directly in a question of tangency, must yet be tested by 
the same principles as those we have examined. 

" If the side of a square be represented by x its area will 
be represented by x 2 . When the side of the square is 
increased by /i, and become x-\-/i y the area will become 
(x+/i) 2 , which is equal to 

x 2 -\-2xh-\-h 2 
While the side has increased by h, the area has increased by 
2xh-\-h 2 . If then we employ h to denote the rate at which 
x increases, 2xh-\-h 2 would have denoted the rate at which 



INTRODUCTION. XXV11 

trie area increased had that rate been uniform ; in which case 
we should have had the following proportion, 
rate of increase of the side : rate of increase of the area 
: : // : 2x7i-\-h~ : : i : 2x+7i 

but since the area of the square increases each instant more 
and more rapidly, the quantity 2x-\-7i is greater than the 
increment which would have resulted had that rate been 
uniform ; and the smaller h is supposed to be, the nearer does 
the increment 2x-\-k approach to that which would have 
resulted, had the rate at which the square was increasing, 
when its side became x, continued uniform. When // is equal 
to zero this ratio becomes that of 

i : 2X 
which is, therefore, the ratio of the rate of increase of the 
side to that of the area of a square when the side is equal 
to x" 

This demonstration is plausible, but will not bear close 
examination. We are told that when h is equal to zero, the 
ratio of the rate of increase of the side of the square to that 
of its area is that of i to 2x. But by hypothesis h repre- 
sents the rate of increase of the side of the square ; if then 
h becomes zero, the side (and of course the area) of the 
square will have no rate of increase, and hence instead of 
rate of increase of the side : rate of increase of the area : : i : 2X 
we shall have 

o:o::i:2i or | = 2,r 
a result that we are already familiar with. 

A COMBINATION OF METHODS. 

Another text-book of very high authority sets forth the 
following method, which seems to be partly derived from 
Lagrange's idea of derived functions. We give the author's 
explanation in his own words : 



XXV111 INTRODUCTION. 

" To explain what is meant by the differential of a quantity 
or function, let us take the simple expression 

u=ax 2 (i) 

in which u is a function of x. Suppose x to be increased by 
another variable h j the original function then becomes 
a{x+ft) 2 j calling the new state of the function u' we have 

u f =a(x -\-/i) 2 = ax 2 + 2axh-\-ah 2 
From this subtracting equation (i) member from member 
we have 

u' —u = 2ax7i -{-ah 2 
The second member of this equation is the difference be- 
tween the primitive and new state of the function ax 2 , while 
h is the difference between the two corresponding states of 
the independent variable x. As h is entirely arbitrary, an 
infinite number of values may be assigned to it. Let one 
of these values, which is to remain the same while x is inde- 
pendent, be denoted by dx, and called differential of x, to 
distinguish it from all other values of h. This particular 
value being substituted in equation (i) gives for the corres- 
ponding difference between the two states of u or x 2 

u ! —u = 2ax dx+a(dx) 2 (2) 

Now the first term of this particidar difference is called the dif- 
ferential of u, and is written 

du = 2ax . dx 
The coefficient (2 ax) of the differential of x, in this expres- 
sion, is called the differential coefficient of the funciio?i u, and 
is evidently obtained by dividing the differential of the func- 
tion by the differential of the variable, and is in general 
written " 

du 

&=*** (3) 

Now will any inquiring student be satisfied with this 
" explanation ? " Will he infer from it anything of the nature 



INTRODUCTION. XXIX 

and office of a differential, and what is its philosophical 
relation to the function to which it belongs ? 

But perhaps this is not intended as a full explanation, for 
the author proceeds : 

" Resuming this expression 

id — u — 2axh-\-ah 2 

and dividing by //, we have 

u f — u 

— ■; — —2ax-\-ah 

In the first member of this equation the denominator is the 
variable increment of the variable x, and the numerator is 
the corresponding increment of the function uj the second 
member is then the value of the ratio of these two incre- 
ments. As h is diminished, this value diminishes and be- 
comes nearer and nearer equal to 2ax, and finally when^=o 
it becomes equal to 2ax. From this we see, that as these 
increments decrease, their ratio approaches nearer and nearer 
to the expression 2ax, and that by giving to h very small 
values, this ratio may be made to differ from 2ax by as small 
a quantity as we please. This expression is then properly, 
the limit of this ratio-, and is at once obtained from the value 
of the ratio by making the increment h=o. It will also be 
seen that this limit is precisely the same expression as the one 
which we have called the differential coefficient of the func- 
tion u" 

Now we have here a term arbitrarily selected without 
explanation or apparent reason, in which h has a fixed value ; 
and this term is called the differential of the function. But 
afterwards in order to find the value of this term it becomes 
necessary to reduce h to zero. The question arises here, if 
h must be zero in the one case why must it not be zero in the 
other ? It will not answer to say that 2 ax, the differential 
coefficient, does not contain h or dx, and is therefore not 
affected by its value ; because dx does occur in the first mein- 



XXX INTRODUCTION. 

ber of equation (3) as its denominator, and hence has to do 
with the value of 2ax. We have then 2ax=—j-, where dx\s 

u f — u 
a particular value of k, and 2ax = — j — where h is equal to 

zero. If the case be so that the value of the fraction 

du u — u . . 

-T- or — t — is independent of that of the denominator, the 

author nowhere tells us why it is so. 

The object of the author in this setting forth of his method 
is evidently to avoid the apparent absurdity of making dx=o 
while it performs so important an office in his subsequent 
analysis; thus escaping one of the inconsistencies of the 
method of limits. But as he practically uses that method in 
the first part of his work, there seems to be after all some 
confusion of ideas, and it is difficult to regard dx as having 
a fixed value, while its representative h is continually re- 
duced to zero. The author himself seems to have become 
wearied with this indirect and misty method, and when he 
comes to the practical application of the Calculus to Geom- 
etry, comes squarely down on to the infinitesimal system 
which, with all its inconsistencies, is far more direct and 
fruitful in its results than the method of limits. 

The most striking circumstance in connection with every 
modification of the method of limits is, that the value of the 
differential coefficient is the objective point sought after. 
Whether this is obtained by a sound, logical demonstration 
or not, it is the only thing found which has a real value, of 
which we can form a definite conception. Now the differ- 
ential itself is quite as important as the differential coeffi- 
cient. It is true we do not regard its actual measured value, 
but we do regard its relative value as compared with that of 
other differentials ; and for this purpose we need some value 
that the mind can grasp and upon which the imagination can 



INTRODUCTION. XXXI 

rest with satisfaction. But none of the systems we have 
examined present the idea of a differential as consisting of 
any such quantity. 

The infinitesimal system does indeed profess to give a sort 
of value to the differential. It is the " last value of the 
variable before it becomes zero," or " the difference between 
two consecutive values of the variable " — words that con- 
vey no more definite idea of quantity than a sort of attenu- 
ated essence of one about to vanish ; while the system of 
limits leaves us nothing whatever but the ghosts of those 
that have departed entirely. 

Who would not desire to be relieved from the constant 
strain upon the imagination, and the severe draft on the 
faith required to attain the results of the calculus by such 
feeble means ? What a relief to have placed in our grasp a 
principle that has substance and vitality; that is adapted to 
our conceptions and meets the demands of our reason, whose 
meaning is not dim nor doubtful, but clear as the noonday 
sun, shining by the light of its own self-evident truth. 

We have said that a fundamental error runs through all 
the systems of infinitesimals and limits, arising from the 
method of approaching the subject. To understand clearly 
what this error is we must have a clear conception of the 
true nature of a differential, and of the symbol which rep- 
resents it. As we have already stated, it is truly defined by 
Prof, Loomis as" the rate of variation of a function or of any 
variable quantity" and further, " by the rate of increase at 
any instant we understand what would have been the absolute 
increase if this increase had been uniform." And the dif- 
ferential coefficient is the ratio between the rate of varia- 
tion of any variable and the consequent rate of variation of 
the function into which it enters. Now this ratio is really 
what is sought after in both the systems that we have exam- 
ined. In the system of Leibnitz the infinitesimal increments 



XXX11 INTRODUCTION. 

represent the rates of increase, and in the method of limits 
the ratio of the rates is obtained from that of the actual 
increments by reducing them to their vanishing point. Not 
that the authors of these systems were conscious of any 
such meaning in their methods, but this was, nevertheless, 
the real, though unrecognized, philosophy on which those 
methods were based/ Now the error which gave rise to all 
the absurdities, sophisms and obscurities of their system was 
this — they endeavored to arrive at the ratio of the rates of 
change by means of the actual changes. That is, they gave to 
the variable an increment, and to the function a correspond- 
ing one, and from these attempted to derive what is really the 
ratio of the rates, or the differential coefficient. This can- 
not logically be done, except in the case of uniform varia- 
tion ; for in all other cases the rate changes as the value of 
the function changes ; so that before the rate can be meas- 
ured by any actual change, it will itself have changed. 
Take a familiar illustration. It is a well-established fact in 
Natural Philosophy, that the velocity of a body falling in 
vacuo to the earth cannot possibly be measured for any one 
instant by the actual movement of the body subsequent to 
that instant, for no such subsequent movement will be made 
with the same velocity. Now if no actual change can rep- 
resent a variable rate of change, the ratio of the actual 
changes cannot truly represent the ratio of the rates how- 
ever small they may be made. 

It is this effort to do what is, in the nature of things, im- 
possible, that has introduced all the difficulties, enigmas and 
mysteries that have beset the differential calculus from the 
beginning. Now these are as unnecessary as they are 
objectionable. The true principles of the science are as 
clear and consonant with reason as the elements of Euclid, 
and the science itself flows from them as directly as the 
light from the sun. Not only so, but while the methods we. 



INTRODUCTION. XXX111 

have examined have produced a study hard and unattractive, 
consisting almost entirely of manipulations of the mere 
machinery of analysis, the subject is really full of beauty, 
abounding in ideas of the most novel and interesting kind, 
and furnishing a field for the exercise of the imagination 
that will tax all its powers — it is, in fact, the poetry of 
mathematics. 



THE TRUE METHOD OF NEWTON. 

The method of arriving at the differential coefficient by 
means of the ultimate ratios of the increments, or, in other 
words, the method of limits, has generally been ascribed to . 
Sir Isaac Newton; but this is evidently an error. The 
theory on which that method is founded is certainly his, and 
it is but just that he should be held responsible for the re- 
sults that legitimately flow from it. But it is not the theory 
on which he formed his method of fluxions. That is con- 
tained in the second lemma of the second book of his Prin- 
cipia. In a scholium to that lemma he says : " In a letter 
of mine to Mr. J. Collins, dated Dec. 10, 1672, having 
described a method of tangents — which at that time was 
made public, I subjoined these words. This is one particular 
or rather corollary, of a general method, which extends itself, 
without any troublesome calculation, not only to the drawing of 
tangents to any curved lines, whether geometrical or mechanical, 
or any how resolving other abstruse kinds of problems about the 
crookedness [curvature] areas, lengths, centers of gravity of 
cicrves, etc. , nor is it limited to equations which are free from 
surd quantities . This method I have i7iterwoven with that other 
of working equations, by reducing them to infinite series. So 
far that letter. And these last words relate to a treatise I 
composed on that subject in the year 1671. The founds 



XXXIV INTRODUCTION. 

tion of that general method is contained in the preceding 
lemma. " 

Here it is distinctly stated by Newton himself that he had 
invented a general method which was applicable not only to 
the drawing of tangents, but to all the higher and more del- 
icate problems which appear in the differential calculus, and 
that this general method has the lemma in question for its 

FOUNDATION. 

We have then but to examine this lemma to ascertain the 
real basis on which the " method of Newton " was con- 
structed. For this purpose we give the lemma in the author's 
own words. 

LEMMA II. 

" The moment of any genitum is equal to the moments of each 
of the generating sides drawn into the indices of the powers of 
those sides, and into their coefficients continually. 

" I call any quantity a genitum which is not made by the 
addition or subduction of divers parts, but is generated or 
produced in arithmetic by the multiplication, division or ex- 
traction of the root of any terms whatsoever; in geometry 
by the invention of contents and sides, or the extremes and 
means of proportionals. Quantities of this kind are pro- 
ducts, quotients, roots, rectangles squares, cubes, square 
and cubic sides and the like. 

" These quantities I here consider as variable and inde- 
termined, and increasing or decreasing as it were by a per- 
petual motion or flux ; and I understand their momentane- 
ous increments or decrements by the name of moments ; so 
that the increments may be esteemed as additive or affirm- 
ative moments, and the decrements as subducted or nega- 
tive ones. But take care not to look upon finite particles as 
such. Finite particles are not moments, but the very quan- 
tities generated by the moments. We are to conceive them 



INTRODUCTION. XXXV 

as the just nascent principles of finite magnitudes. Nor do 
we in this lemma regard the magnitudes of the moments, 
but their first proportion as nascent. It will be the same 
thing, if, instead of moments, we use either the velocities of 
the increments and decrements (which may be called the 
motions, mutations and fluxions of quantities), or any finite 
quantities proportional to those velocities. The coefficient 
of any generating side is the quantity which arises by 
applying the genitum to that side. 

" Wherefore the sense of the lemma is, that if the mo- 
ments of any quantities A, E, C, etc., increasing or decreas- 
ing by a perpetual flux or the velocities of the mutations 
which are proportional to them, be called a, b,c, etc., the 
moment or mutation of the generated rectangle AB will be 
#B-f bA ; the moment of the generated content ABC will be 
aBC-\-bAC-{-cAB; and the moments of the generated pow- 
ers A 2 , A 3 , A 4 , A*, A* A*, A* A" 1 , A" 2 , A"* will be 
2aA, 3#A 3 , 4^A 3 , %aA 2 , f^A 2 , \aA 3 , f#A 3 , — aA~ 2 , 

■ _3. 

— 2aA~ s , — \aA 2 respectively; and in general that the 

n n—m 

moment of any power A- 71 will be ^flA m . Also that the 

moment of the generated quantity A 2 B will be 2aAB+bA 2 ; 

the moment of the generated quantity A 3 B 4 C 2 will be 

3^A 2 B 4 C 2 +4^A 3 B 3 C 2 + 2^A 3 B 4 C; and the moment of the 

A 3 
generated quantity ^j, or A 3 B~ 2 , will be 3^A 2 B -2 — 

2#A 3 B -3 , and so on. The lemma is thus demonstrated. 

" Case i. Any rectangle, as AB, augmented by a perpet- 
ual flux, when as yet there wanted of both sides A and B, 
half the moments \a and ^b, was A—\a into B— \b, or 
KB—\aB—\bA-\-\ab; but as soon as the sides A and B are 
augmented by the other half moments, the rectangle be- 
comes A+^a into B+-p, or AB - r \aB -\-\b A -\-^ab. From 



XXXVI INTRODUCTION. 

this rectangle subduct the former rectangle, and there re- 
mains the excess aB-\-bA. Therefore with the whole incre- 
ments a and b of the sides, the increment aB-j-bA of the 
rectangle is generated Q. E. D." 

" Case 2. Suppose AB always equal to G, and then the 
moment of the constant ABC or GC (by case i) will be 
g-C+rG, that is (putting AB and aB+bA for G and g) 
aBC-+bAC+cAB. And the reasoning is the same for con- 
tents under ever so many sides. Q. E. D." 

It is unnecessary to quote the demonstrations of the other 
cases, as they all flow naturally and logically from these which 
form the key to the whole system. 

We must concede that this demonstration is not as clear 
and complete as could be desired. Let us, however, 
endeavor to extract from it the real, though perhaps some- 
what vague conception of the subject which occupied the 
mind of Newton. It is to be remarked, however, that the 
doctrine of limits is nowhere hinted at, but the results are 
direct, positive and substantial. 

The first question suggested by the lemma is, what is 
really meant by the term " moment." It might at first seem 
that the " moments " of Newton were in fact the same thing 
as the differentials of Leibnitz, for he speaks of them as 
something (though not finite quantities) to be added or sub- 
tracted. But a very little examination of the lemma will 
dispel the notion. Their magnitudes are not to be regarded. 
But the magnitudes of the differentials of Leibnitz are to be 
regarded as infinitely small. Again, " finite particles " are 
not " moments," but the " very quantities generated by the 
moments." Now the differentials of Leibnitz never gener- 
ate anything; they are the infinitesimal remains of incre- 
ments that have been added and then taken away. Again, 
moments are the "nascerit principles of finite magnitudes." 
But the " principles " which generate " finite magnitudes " 



INTRODUCTION. XXXV11 

or increments can be nothing else than the laws which con- 
trol the changes in the " genitum ;" that is, the rate of 
change. This interpretation is confirmed by the further 
statement that we may use instead of them " the velocities " 
or any finite quantities proportional thereto. Hence we 
infer that a, b, c, which are called moments, are intended as 
symbols to represent the rates of change, being finite quan- 
tities proportional to those rates, and as the quantities 
A, B, C, etc., are increasing or decreasing by a "perpetual 
flux," that is by a uniform rate of change, the actual incre- 
ments or decrements a, b, c will represent those rates. So 
that the difference between A—\a and A-\-\a (equal to d) 
represents the rate of increase of A, and the difference be- 
tween B — \b and B+^b (equal to b) accruing during the 
same time represents the corresponding ra.te of increase of 
B ; and the ratio of a to b represents the ratio of those rates 
whatever may be their magnitude as symbols. But while 
these symbols or suppositive increments (being produced 
at a uniform rate) represent the respective rates of increase 
of A and B, we are told that the corresponding increment 
of their product (aB-rbA) represents the "moment" or rate 
of increase of their product. Now as the product does not 
increase at a uniform rate, it becomes a question why this 
increment should represent the rate of increase of AB. This 
is probably one of those cases in which the intuitive per- 
ceptions of Newton seized the true result without stopping 
to elaborate the intermediate steps. At all events he has 
here presented the only true key that will completely unlock 
the calculus ; and this key we shall in due time apply to that 
purpose. 



XXXVlll INTRODUCTION. 



[The following letter from the author to a correspondent contains a 
more elaborate explanation of some of the principles of the calculus than 
will be found in the text of this work. It was written to meet certain 
objections, which will appear in the progress of the letter, and is inserted 
here in order to meet the same objections should they arise in the minds 
of others.] 

Chicago, Jan. 22, 1875. 
Dear Sir : You say that you are in doubt as to what 
meaning I attach to the statement: "This law is derived, 
not from any actual change, but from the conditions con- 
tained in the algebraic formula by which the variable func- 
tion is expressed." My meaning is, that in any variable 
function, as for instance ax—bx' z the rate of variation, or, in 
other words, the law of change in the function is to be de- 
rived, not from giving an actual increment to x and a corres- 
ponding one to the function, but from the conditions expressed 
in the formula; that is from the expressed relation of x to 
its function; and the expression of this law, derived from the 
formula is — the change that would take place in the function, 
arising from the rate of change in x, if the change in the func- 
tion were to continue uniform. To find this suppositive uni- 
form change in the function is one of the problems of this 
work. We will take however a very simple case for illustra- 
tion. Suppose we have the function ax in which x is chang- 
ing at a variable rate. The law of change in this function 
is derived from the simple condition expressed in the for- 
mula, viz., that as the value of the function is always a times 
that of x, the rate of change in the function will always be 
a times that of xj and whatever uniform change may be 
given to x to express its rate of change, the corresponding 
uniform change in the function will be a times that of x; and 



INTRODUCTION. XXXIX 

this is the " law of change derived from the conditions con- 
tained in the algebraic formula ax" — hence d(ax)=adx. 

Having thus expressed my meaning, permit me to reply 
to some expressions in your letter which I will quote out of. 
their connection, but not, I think, so as to change their 
meaning. You say: "The amount of change in any par- 
ticular case will be determined by the law," etc. I under- 
stand you to mean that the amount of change for any unit 
of time will be determined by the law governing the change 
— that is, by the rate of change. This is true for a uniform 
change but not for a variable one. For, suppose two quan- 
tities are changing, at a variable rate, in such a manner that 
at a certain instant, the rate of each is the same; then the 
law of change at that instant is the same in both, while the 
actual amount of change in any unit of time in each quantity 
may be as different as possible from the other. Hence the 
amount of a variable change will not be determined for any 
unit of time by the law or rate of change. 

Again you say, in reference to my statement already 
quoted : " I see no occasion for the remark referred to unless 
it mean that there is no such actual change as the supposi- 
tive change symbolized by dx ; a very different statement 
from the one made." 

Now it seems to me that we are using the same word in 
different senses. I mean by a suppositive change in the case 
of a variable rate, an ideal, or (if you please) fictitious or 
hypothetical one, which is not only different from any actual 
change but different from what any actual change can be. 
Hence then, of course, there can be " no such actual change 
as the suppositive one symbolized by dx ;" and I have re- 
peatedly stated that the change symbolized by dx was not 
an actual change at all. If, for example, I say that a body 
is falling freely to the earth at the rate of fifty feet in one 
second, this fifty feet is wholly an ideal change of position, 



xl INTRODUCTION. 

or increment of space ; and I would never be understood as 
asserting that it was an actual increment of the space passed 
over by the falling body in one second ; nor that any fifty 
feet would be uniformly described; and yet my assertion 
would mean fifty feet passed over at a uniform rate; but it 
would be a " supfiositive" fifty feet, and would be passed over 
in one second oji the supposition that the rate continued uni- 
form for that time : but that never does or can take place in 
reality. 

Again you say: "Admitting your idea of variables as in 
a state of change, it still seems to me that in the calculus, 
we are concerned with the changes which result from that 
state, not with the state itself; that dx is a symbol not of a 
state, but of a certain amount of change that would accord- 
ing to your definition occur in a unit of time, were the 
change at a particular point, or rather any point continued 
uniformly for that unit, and that if dx is an entity there must 
be an actual change as its basis." 

Now in the calculus it is exactly the state of change, and 
not the actual change with which we are concerned. The 
latter belongs more properly to Natural Philosophy. It is 
true that dx is the symbol of a certain amount of supposi- 
tive change (not real except the rate be uniform), and this 
symbol expresses the state of change, and not an actual nor 
(where the rate is variable) even a possible one. But be- 
cause the change is ideal, suppositive or hypothetical, and 
not actual nor possible, it does not follow that it is not an 
entity. The ideal change is the symbol of the rate, and the 
rate though not a simple quantity, but a relation between two 
other quantities (viz., change, ard the time occupied by it), 
and, therefore, represented by a symbol is still an entity, for 
anything that exists is an an entity. And although rate is 
not change, it does not follow that it is based on " no 
change;" it is something wholly different from change, 
although represented by it as a symbol. 



INTRODUCTION' xll 

You ask in reference to signs : " Do we not in Trigonom- 
etry obtain our general formulas as though the functions of 
the angle were all plus, and when applying them make such 
changes as the signs of particular values require?" 

In Trigonometry we consider all the functions as essenti- 
ally positive in the first quadrant, but outside of that some 
of them are essentially negative, whatever may be the sign 
prefixed. We must distinguish in the calculus between the 
signs plus and minus, and the terms positive and negative. 
The former denote the relation of one quantity to another, 
the latter the essential character of the quantity. 

I now come to the last point mentioned in your letter. It 
is a vital one, and I wish in discussing it to make my argu- 
ment such, that taken in connection with the demonstration 
in the book itself, it will be complete and exhaustive. 
Hence I ohall be obliged to repeat some things that I have 
said in order that the continuity of the argument may not 
be broken. 

You say, " I do concede that you have logically obtained 
the differential of the product of two variables upon the 
assumed hypothesis, viz., that the variables changed uniformly; 
that if they do not change uniformly, then by the method 
you have adopted du would not be the change in u corres- 
ponding to contemporary states of x and y for x and y would 
not be contemporaneous. " 

The question, therefore, is, will the demonstration which 
I have given, and which you concede to be logical (and, of 
course, conclusive) for obtaining the differential of a pro- 
duct of two variables when their rate of change is constant, 
be also true when their rate of change is variable ? 

First. What do we mean by rote of change ? Rate is a 
complex idea. Its elements are change, and the time con- 
sumed by it. It is, in fact, the relation between these two. 
It is an entity because it exists. It is a quantity because it 



xlii INTRODUCTION* 

can be increased, diminished and measured ; but it is a very 
different kind of quantity from either time or change. These 
latter are quantities of which the idea is objective, presented 
externally to the mind; while rate is a quantity of which 
the idea is subjective, originating in the mind and only capa- 
ble of being represented by a symbol. This symbol is of 
course arbitrary. Sometimes it is the time (the change be- 
ing known) which increases as the rate diminishes and vice 
versa. But in the Calculus the symbol is the ehange, which 
increases and decreases with the rate ; and as this symbol is 
only used to compare one rate with anather, we need not 
regard the element of time farther than to make the symbol- 
ical changes simultaneous. By thus leaving out the element 
of time the idea of rate has become so far identified with 
that of its symbol, that we often fail to make any distinction 
between them ; but they are as different from each other as 
an angle from its sine or tangent. 

Second. The change required to be the symbol of a rate 
must be a uniform one. Thus, if the rate of a falling body 
is said to be fifty feet in one second, it is meant that it would 
pass over that space in one second of time with a uniform 
motion if the existing rate were not disturbed for that time. 
No change whatever except a uniform one can be a symbol 
by which a rate will be correctly expressed. 

Third. To apply the idea of rate to an actual change 
which we will suppose to be a variable one, let us again take 
the case of a falling body, which at the moment in question 
has a velocity of fifty feet in one second. Then rate being, 
as I have said, the relation between time and a uniform 
change, it follows that in this case the rate is the relation of 
one second of time to fifty feet of space uniformly described 
in that time, and this relation is the law of the motion at 
that instant, and unless interrupted by some external cause 
would continue to govern the motion indefinitely. Now in 



INTRODUCTION. xliil 

this case, this law applies to no other instant of time nor 
point of space. For suppose a body to be projected upward 
with such a velocity that at the moment when the velocity 
of the falling body had increased to fifty feet in one second, 
that of the rising body should have diminished to the same 
rate. Then at that moment the law governing the motions of 
both these bodies would be exactly the same. If the bodies 
are equal in weight and should both strike an obstacle, the 
effect of the impact would be the same in both cases, and 
also the same as if their motions were uniform at that rate ; and 
yet they do not move at the same rate during any time at all. 
The existence of a rate then in a variable change is instan- 
taneous j it occupies no time; no aciical change takes place 
in accordance with the rate ; and yet it is the same, its effects 
are the same, and its relations are the same at the instant as 
though it were not changing at all. 

Fourth. In obtaining then the rate of change in a pro- 
duct of two variables, we have only to consider their rates 
as they exist at that instant, and the result will be the same 
whether those rates are constant or not; for we are not con- 
cerned with the rate at any other time either before or after 
that instant. 

It may be asked if a rate occupies no time and involves 
no actual uniform change, why do I make use of such uni- 
form changes in obtaining the rate of a product ? I reply, 
the uniform changes used for this purpose are not real, but 
suppositive or fictitious changes, which are symbols of the 
rates existing at the moment in the variables whose product 
is under investigation. The use therefore of such uniform 
suppositive changes in obtaining the rate of the product is 
not at all inconsistent with the proposition that the rate in a 
variable change does not require time non actual uniform 
change for its existence, (viz, its occurrence or taking place). 
Fifth. I have said that the change which symbolizes the 



xlvi INTRODUCTION. 

grasp and the mind comprehend ; and it is such as these 
that I propose to substitute for the incomprehensible infini- 
tessimal or the equally incomprehensible quotient of nothing 
divided by nothing. I am, etc. 

The following outline of the argument in the foregoing 
letter may be of service in drawing attention to the chain of 
reasoning : 

First. A differential is a rate of change. 

Second. A rate is a subjective idea and can only be ex- 
pressed by a symbol. 

Third. That symbol in the Calculus must be a uniform 
change, which, in a variable rate, is a fictitious one. 

Fourth. In a variable rate one such symbol can apply to 
but one value, and hence in such a rate that one value requires no 
appreciable time for its existence nor actual uniform change. 

Fifth. In comparing rates we compare their symbols, and 
in estimating them we estimate the symbols. 

Sixth. Hence all the relations, effects and values of rates 
are at any one moment the same as if they were uniform. 

Seventh. Hence a demonstration applicable to uniform 
rates wiil apply at any one moment to variable rates also, for 
the symbols, which alone are used in the demonstration, and 
by which the rates are expressed, compared and estimated, 
are precisely the same uniform changes, whether the rates 
are uniform or variable. 

Eighth. If a variable is changing at a variable rate, 
while that fact will not affect the expression for the differen- 
tial of its function, it will very sensibly affect the differential 
of the rate of the function, which is its second differential; 
and it is here and not in the first differential that we are to 
consider whether the rate of the variable is uniform or not. 
Thus while d(xy) is always ydx+xdy, yet if dy is variable, 
d 2 xy = 2dxdy+xd 2 y instead of 2dxdy as when the rates of 
both are constant. 



PART I. 



Differential Calculus. 



Differential Calculus. 



SECTION I. 

DEFINITIONS AND FIRST PRINCIPLES. 
VARIABLES. 

(I) Two classes of quantities are considered in the dif- 
ferential calculus, namely, variables and constants. 

Variables are quantities that are in a state of change ; that 
is, their values are in an increasing or decreasing condition ; 
such, for example, as the quantity of water in a vessel which 
is being filled or emptied by a continuous stream ; or as the 
force of attraction which increases or diminishes as the 
attracting bodies approach or recede from each other ; or 
as the space between these same bodies while they are mov-' 
ing. They are, in the differential calculus, not merely quan- 
tities subject to change, or to which different values may be 
assigned ; but quantities in which the change is stipposed to be 
actually occurring at the moment when they become the sub- 
ject of the analysis. It is their actual condition and not their 
attributes or qualities that are referred to in this definition. 
Take for example the space passed over by a falling body. 
That space is a variable, not because it may or does have 
different values, but because its value is constantly chang- 

4 



50 



DIFFERENTIAL CALCULUS. 



ingy or is in a slate of change. It is this state, and not any 
actual change, that is the peculiar subject of the transcen- 
dental analysis. 

RATE OF VARIATION. 

(2) Rate of variation is the relation between the change of 
a variable and the time occupied by the change. Being a 
relation and not a simple quantity, it can only be represented 
by a symbol, which is, a uniform change in a given unit of time. 
If the rate is constant, then the actual change is the true sym- 
bol, but if it is variable, then the change must be a supposi- 
tive one — that is, one that would take place in the same unit of 
time if it were to continue uniform. Thus in the case of a fall- 
ing body; it is said, the velocity at a certain moment is so 
many feet in one second. It is not meant that the body actu- 
ally falls through that distance in one second, nor any dis- 
tance whatever at that rate, but that it would fall so far if the 
velocity existing at that moment were to continue uniform for 
one second. The velocity belongs to that one moment, and 
that one position only. At the very next point above and 
below this position the velocity is different, and hence no 
actual movement, however small in respect to space and time, 
can pcssibly represent it. This will be seen at once if we 
consider what velocity is. It is not of itself a quantity, but 
a relation, which refers, not to the place, but to the condition 
of the body in respect to the motion — that is, to the degree 
or intensity of the state of motion in which the body is. 

So it is with all variables. The rate of change refers to the 
intensity with which the change is going on, and if it is not 
uniform it cannot possibly represent the rate, for it lacks the 
essential element of the required symbol. The latter must 
therefore be obtained from the law which governs the change 
and not from the change itself. Hence instead of giving to a 
function an actual increment for the sake of obtaining its rate of 



DEFINITIONS AND FIRST PRINCIPLES. 5 1 

increase at any moment, we examine the law which governs 
the change ; and the expression of this law is the change that 
would take place in a it nit of time if the rate were to continue 
uniform j and this is the measure of the rate. 

Note. — It must be remarked that the ide.s of time, motion and velocity, attached 
to the ordinary meaning of these words, have no place in the abstract science of the 
differential calculus. The term motion and velocity are used in this article merely to 
illustrate the meaning of the term "rate" It is true that velocity is a rate — the 
rate of motion. But many other things beside motion have a rate ; such as the varia- 
tion of light, heat, magnetism, force, anything which increases or diminishes by the 
operation of a prescribed law ; and the calculus is applicable to all such subjects where 
the conditions can be expressed analytically. 

The idea of time in its absolute sense is also foreign to the calculus. The term 
•* unit of time" in the definition does not refer to any specific portion of time, it may 
be great or small ; its value does not enter into the calculation, and hence this system 
does not in any wise invade the domain of natural philosophy. All that the abstract 
science of the calculus has to do with time, is confined to the simple condition that the 
suppositive changes in the value of the variable and its function, which symbolize their 
rates of change, shall be simtdtaneous. And this is no more than all systems of the 
alculus require for th e actual changes which are supposed to be made in the same 
quantities. 

In the application of the calculus to Geometry the idea of motion is in some sort 
introduced ; but not, however, in its philosophical sense as having an absolute value. 
Geometrical magnitudes are supposed to be generated by the movement of their ele- 
ments. Thus a line is generated by the flowing of a point, a surface by a line, and a 
solid by a surface ; and this conception is used to determine the proportion of magni- 
tudes, by comparing the rates at which they are generated instead of comparing the 
magnitudes themselves with each other. This idea of the generation of magnitudes 
by means of their elements is not new in mathematics. It is one of the seminal 
ideas of the Cartesian system ; and though in this work it is certainly made more 
prominent than it has usually been, and more prolific in results, it is not therefore out 
of place. 

DIFFERENTIALS. 

(3) The differential of a variable, or function, is its rate of 
change or variation, symbolically expressed by the suppositive 
change that would take place at that rate. 

If the variable is essentially positive and increasing, or neg- 
ative and decreasing, its differential will be essentially posi- 
tive. If it is essentially negative and increasing, or positive 
and decreasing, its differential will be essentially negative. 
Thus, if we consider a northern latitude positive and a 



52 DIFFERENTIAL CALCULUS. 

southern negative, a vessel will have a positive rate (or dif- 
ferential) of progress if her northern latitude is increasing 
or her southern latitude is decreasing; and vice versa her 
rate of progress will be negative. 

The notation used to designate the rate of variation or dif- 
ferential is the letter d placed before the variable, whose 
rate is required. Thus the differential of x is written dx. 
If the variable is a component expression such as x 2 +ay, 
the differential would be written d (x 2 -\-ay). Variables are 
themselves indicated by the last letters of the alphabet. 

Note. — I use the nomenclature and notation of Leibnitz, not because there is any- 
actual necessity for so doing, but because their use has become so general not only in 
the system of Leibnitz, but also in other systems, that it seems to have become fixed, 
without much regard for their original derivation and meaning ; and hence a change 
in that respect would appear like an unnecessary innovation. 

The term "fluxion'' is more truly significant of the true principles of the science 
then the term u differential," and the symbol " d " is no better than some other would 
be ; but it is just as good as any, and the use of it involves no inconsistency with the 
principles on which the system is based. 

CONSTANTS. 

(4) The other class of quantities which enter into the 
transcendental analysis is that of constants. These are sup- 
posed to have a fixed value, although it is not always neces- 
sary that this value should be known or given. In equa- 
tions the constant quantities express the conditions of the 
proposition, and while they are generally supposed to have 
a given, or, at least, an assignable value, there are many cases 
in which their value must be determined by the solution of 
an equation just as any unknown quantity in algebra is deter- 
mined. This, however, does not make them variables ; their 
value is as much fixed as if it were known at first. The 
solution of an equation is rendered necessary in order to 
make the immediate conditions conform to some ulterior 
conditions imposed upon them. Thus the general equa- 
tions of two circles will determine the curves when the con- 



DEFINITIONS AND FIRST PRINCIPLES. 53 

stams are given ; but if there is an ulterior condition that 
they must be tangent to each other, the constants must be 
made to conform to this condition ; which can only be done 
by an equation from which the necessary values can be 
obtained. This solution does not fix the values of the con- 
stants, but only makes known those values which were fixed 
or rendered certain by the conditions to which they were 
subjected. A constant then is never in a state of variation. 

FUNCTIONS. 

(5) A function of a variable is any algebraic expression whose 
value depends on that of the variable. Thus 

ax 2 — 2x 
is a function of x since its value changes with that of x, sup- 
posing a to be constant. The expression 

ax -{-by 
is a function of x and y [a and b being constant), for it de- 
pends on both x and j' for its value ; and thus we may have 
a function of any number of variables. 

When the expression does not involve an equation, the 
variables are independent of each other ; that is, we may 
assign to any of them any value whatever without regard to 
the values assigned to the rest. But an equation which con- 
tains variables will have at least one dependent on the others 
for its value. Thus in the equation 

y~ -\-bu-ax 
in which x y y and ?/ are variables, we may give arbitrary 
values to any two of them, but the value of the third must 
be determined from the equation. This last is called a func- 
tion of the others and the dependent variable, while the oth- 
ers are called independent variables. When the dependent 
variable stands alone in one member of the equation it is 
called an explicit function of the others, but when combined 



54 DIFFERENTIAL CALCULUS. 

with the others it is called an implicit function. The term 
function, however, applied to the dependent variable is to 
be understood as meaning the representative of the function, 
and not, literally, the function itself. 

Functions are commonly divided into two classes, which 
are distinguished by the manner in which the variables enter 
into them, and are called "Algebraic " and" Transcendental." 

Algebraic functions are those in which the variables are sub- 
jected only to the operations of addition, sicbtraction, multiplica- 
tion, division, and involution or evolution, denoted by constant 
exponents or indices. 

Transcendental functions are those in zuhich the variable is 
either an exponent, logarithm or trigonometrical line, such as a 
sine, tangent, etc. This distinction is not important, however, 
and may be disregarded. 

(6) The fundamental problem of the calculus is to find 
the differential, or rate of change, in a function of a varia- 
ble produced by that of the variable itself 

As the differential or rate of change in a variable is rep- 
resented by a suppositive change, taking place at a uniform 
rate, and that of the function arising from it by a correspond- 
ing suppositive change, these changes (being uniform from 
the beginning)will have a constant ratio independent of their 
value. Hence the differential of the variable is always a factor 
of the differential of its function. The other factor, that is, 
the ratio between the differential of the variable and that of 
its function, is called the differential coefficient of the function. 
Since this ratio is not affected by the value of the suppositive 
change representing the differential of the variable, this dif- 
ferential is indicated by an indeterminate symbol, and the 
differential of the function becomes a function of that symbol. 

The differential of a function is obtained by a process 
called differentiation, and the differential coefficient is 
obtained by dividing the differential of the function by that 



DEFINITIONS AND FIRST PRINCIPLES. 55 

of the variable ; so that the differential coefficient of ax 2 — bx 

would be 

d(ax 2 — l?x) 

dx 

If we represent the function by u we have 
u = ax 2 — bx 
and 

-j x = differential coefficient, which 
we can obtain as soon as we know how to find the differen- 
tial of ax 2 — bx. 

(7) If we have an equation containing variables in each 
member, since the two members are always equal, their rates 
of change are also equal : hence we may differentiate 
each member as a separate function, and place the results 
equal to each other. If the equation contains more than 
one variable, one of them will be dependent and the value 
of its differential will depend on the values of the other 
variables and their differentials. Either of the variables 
may be taken as the dependent one, and it will then repre- 
sent a function of the rest. 

If the differential of a function of two or more variables 
be taken with reference to one only, and then divided by its 
differential, the result will be the differential coefficient for 
that variable, and all the rest must be treated as constants 
for that coefficient ; and the function as a function of that 
variable j for a differential coefficient can exist only between a 
single variable and its function. 

If we wish to indicate a function of any variable, as x, 
without giving it any particular form, for the purpose of 
demonstrating some general truth applicable to all forms, 
we use the expression F (x), which means any function 
depending on x for its value. If it is a function of two or 
more variables, the expression is F (x,y), or F (x,y, z), and 
similarly for a greater number of variables. 



SECTION II, 



DIFFERENTIA TION OF FUNCTIONS. 

Proposition I. 

(8) To find the sign with which the differential of the varia^ 
ble must enter that of the function to which it belongs. 

Among the characteristics of quantity as used in the cal- 
culus, is that of being essentially positive or negative, ac- 
cording as it lies on one side or the other of the zero point. 
If, for instance, the value is reckoned toward the negative 
side it is essentially negative independent of the algebraic 
sign that may be prefixed to it. Thus the co-sine of 120 is 
essentially negative, whether it is to be added or subtracted, 
while the sine of the same angle is essentially positive. 
These characteristics are independent of, and wholly dis- 
tinct from, the algebraic signs that may be prefixed to them. 
Hence when quantities enter into a function as variables we 
must inquire what will be the effect of these characteristics 
on the influence which their rate of change will have on 
that of their function, so that we may give to the differen- 
tial of the variable that sign which will produce a rate of 
change in the function in the right direction. 

If, then, a variable, whether intrinsically positive or neg- 
ative, is increasing, its rate of change will affect the rate of 
lis function in the same direction as its value affects the value 

56 



DIFFERENTIATION OF FUNCTIONS. 57 

of the function ; and, hence, having essentially the same 
character (Art. 3) it must have the same sign. 

If it is diminishing, its rate of change will affect that of 
its function in a direction contrary to that in which the value 
of the variable affects the value of its function ; but having 
itself a character contrary to that of the variable (Art. 3) it 
must still have the same sign. 

Let us for instance take the function 

x—y 
in which x is a positive and increasing variable. Now if y 
is increasing its rate will be essentially positive or negative, 
according as y is itself essentially positive or negative (Art. 
3), and must therefore affect the differential or rate of the 
function (to increase or diminish it) in the same direction 
asjy affects its value. If y is diminishing, its rate of change 
will be essentially negative if y is positive, and positive if y 
is negative (Art. 3), and will, therefore, affect the rate of the 
function in a direction contrary to that in which y affects its 
value j but being essentially contrary in its nature to that of 
y, it must enter the differential of the function with the 
same sign in order to produce a contrary effect. 

Thus, whether the variable be intrinsically positive or 
negative, whether it is increasing- or diminishing, the sign 
prefixed to Us differential must in all cases be the same as that 
prefixed to the variable itself. 

ILLUSTRATION. 

If we consider a northern latitude positive and a south- 
ern one negative, and there are two vessels, A and B, sail- 
ing north of the equator, let the latitude of A be represented 
by x and that of B by y. Then the difference of their lat- 
itudes will be x — y. If both are sailing north their rates of 
progress will be positive, and the rate of change in their 



58 DIFFERENTIAL CALCULUS. 

difference of latitude will be the difference of their rates of 
sailing; hence the rate of B, which is the differential of y, 
will have a minus sign. If B is sailing south the difference 
of their latitudes will be still x — y, but since the rate of 
change in this difference is the real sum of their rates of 
sailing, the rate of B being essentially negative (Art. 3) 
must have a minus sign, so that the algebraic difference will 
produce the real sum. 

If B be south of the equator, the difference of their lati- 
tudes will be their real sum, but y being now essentially 
negative must have a minus sign to produce this sum, and 
it will still be expressed by x — y. If B be sailing south (A 
being still sailing north), the rate of B will be negative 
(Art. 3), and since the rate of change in the difference of 
their latitudes is the real sum of their rates of sailing, the 
rate of B must have a minus sign, so that the algebraic dif- 
ference will be the real sum. If B be sailing north its rate 
will be positive (Art. 3), and since in this case the rate ot 
change in the difference of latitudes will be the real differ- 
ence of their rates of sailing, the rate of B must still have a 
minus sign, so that the algebraic difference will correspond 
to the real difference. Hence, if y enter the function with 
a minus sign, its differencial or rate of change will have a 
minus sign, whether y is intrinsically positive or negative, 
or is increasing or diminishing. A similar result would fol- 
low if the sign were plus; the sign of the differential would 
be plus. 

Proposition II. 

"(9) To find the differential of a function consisting of terms 
connected together by the signs plus and minus. 

That is to say, to find the rate of change in the function 
arising from the rates of the variables which enter into it. 

Every term in an algebraic expression may be considered 



DIFFERENTIATION OF FUNCTIONS. 59 

as having a single value, made up, of course, of the respec- 
tive values of the quantities that compose it, and their rela- 
tions to each other, and may, therefore, be expressed by a 
single letter. If a term contain none but constant quanti- 
ties the letter representing it will be considered as a con- 
stant. If it contain variables, the letter representing it will 
be considered as a variable having the same rate of change 
as would arise in the term itself from the rates of change in 
the variables which enter into it. 

It will, therefore, be sufficient to investigate the case of a 
function in which each one of the terms is represented by a 
single letter. 

Let us suppose some of these terms to be variable and 
others constant, and the variables to be changing their values 
at any rate whatever, either uniform or variable, and each 
one independent of the rest. The constants will, of course, 
have no rate of change, and will, therefore, not affect the 
rate of change in the function. 

The differential of each variable will be the suppositive 
uniform change that would take place in it in a unit of time at 
the rate existing at the moment of differentiation ; and the 
differential of the function is the uniform change that would 
take place in it arising from the supposed uniform changes 
in the variables. Now, if we suppose this symbolic or sup- 
positive change to be made in each variable, the correspond- 
ing change in the function will be the algebraic sum of the 
changes in the variables ; and as these are by supposition 
uniform, the change in the function will be uniform also at 
the rate at which it commenced, and will, therefore, be the 
symbol of that rate or the differential of the function. 

For example, let us take the function 
x — y+a — b-\-z-\-c 
in which x, y and z are variables, and a, b and c are con- 
stants, and represent by dx, dy and dz the differentials or 



60 DIFFERENTIAL CALCULUS. 

uniform changes that would take place in x, 7 and z in a 
unit of time from the moment of differentiation. Let us 
also suppose these symbolic changes to take place ; the func- 
tion would then become 

x + dx — y — dy + a — b -{-z-fdz+c 
(Art. 8) and if from this we subtract the primitive function 

we have 

dx — dy-\-dz 

which represents the uniform change in the function aris- 
ing during the same unit of time from the suppositive 
changes in the variables. It is, therefore, the symbol repre- 
senting the corresponding rate of change, or differential of 
the function. Hence , 

d{x — -y-\-a — &-j-z-\-c)=dx — dy-\-dz 
or the differential of a function composed of terms containing 
independent variables, having any rates of change whatever, the 
terms being connected together by the signs plus and minus, is the 
algebraic sum of the differentials of the terms taken separately 
with the same signs. 

Since each of the terms in the case given may represent 
a compound term of any form whatever, it is now necessary 
to examine the method of finding the differential of a single 
term in every form in which it may occur. 

The number of these forms for algebraic terms is limited 
to seven as follows : 

1. A variable multiplied by a constant. 

2. One variable multiplied by another. 

3. A variable divided by a constant. 

4. A constant divided by a variable. 

5. One variable divided by another. 

6. A power of a variable. 

7. A root of a variable. 

These simple forms, or some combination of them, which 
can be dissected and operated by the same rules, constitute 
all that can be assumed by single algebraic terms. 



DIFFERENTIATION OF FUNCTIONS, 6l 

(10) To find the differential of a variable multiplied by a con- 
stant quantity. 

We have seen that 

d (x +y + z + u) —dx +dy -\-dz-\-du 
If we make these variables each equal to x we shall have 

x+y+z + u—^x 
and 

dx -{-dy-\-dz-{-du= z 4 dx 
hence 

d {4x)=4 dx 
As the same reasoning will extend to any number of terms, 
we may make the equation general, and we have 

d (nx)=n dx 
That is, the rate of change of u times x is equal to n times 
the rate of change of x. 

Hence, the differential of a variable, -with a constant coeffi- 
cient, is equal to the differential of the variable multiplied by the 
coefficient. In other words, the coefficient of the variable 
will ateo be the coefficient of its differential. 

EXAMPLES. 

Ex. i. What is the differential of abz? — Ans. abdz. 

Ex. 2. What is the differential of b 2 y? — Ans. b 2 dy. 

Ex. 3. What is the differential of ax -\-cy ? — Ans. adx+cdy. 

Ex. 4. What is the differential of x — by? — Ans. 

Ex. 5. What is the differential of (a+b) x? — Ans. 

Ex. 6. What is the differential of (c—d)y? — Ans. 

Ex. 7. What is the differential of ax+by+czt — Ans. 

Ex. 8. What is the differential of b 2 u-\-c 2 z ? — Ans. 

Ex. 9. What is the differential of a 2 bx+c 2 dy? — Ans. 

Ex. 10. What is the differential of a 2 y — b 2 x? — Ans. 

Ex. 11. What is the differential of b (ay— ex) ? — Ans. 

Ex. 12. What is the differential of c 2 (bx-\-az) ? — Ans. 



62 DIFFERENTIAL CALCULUS 



LEMMA. 



(II) If two variables are increasing at a uniform rate, their 
rectangle will be increasing at an accelerated rate, but the 
acceleration will be constant. 

Let x and y be increasing uniformly at rates represented 
by dx and dy. Then dx and dy will be the actual increments 
of x and y in a unit of time, and at the end of m such units 
xy will have become 

(x + mdx) {y-\- mdy) =xy-\r mydx + mxdy + ;;/ 2 dxdy ( i ) 

In one more unit of time we shall have 
(x + {in + 1 )dx) ( y + ( tn + 1 )dy) = xy-\- (in + 1 )ydx + {in + 1 )xdy 
+ {m + 1) 2 dxdy (2) 

In still another unit of time we shall have 
(x J r{m + 2)dx) {y + {m-\-2)dy)=xy-{- {in-j-2)ydx + {m + 2)xdy 
-\-{iii-\-2) 2 dxdy (3) 

Subtracting the second member of (1) from that of (2) we 
have 

ydx-\-xdy-\-(2in + i)dxdy (4) 

and subtracting the second member of (2) from that of (3) 
we have 

ydx-}-xdy-\-(2in + 3)dxdy (5) 

and subtracting (4) from (5) we have 

2 dxdy (6) 

Now the expression (4) is the increment of the product 
arising from the uniform increments of the variable factors 
during one unit of time, and expression (5) is the increment 
of the product during the next equal unit of time arising from 
the next equal uniform increments of the variables. These 
increments of the product may therefore be taken to represent 
its successive mean rates of increase arising from the uniform 
increase of the variable factors during two equal successive 
units of time ; and the expression (6), which is the differ- 
ence between these rates will represent their acceleration^ 



DIFFERENTIATION OF FUNCTIONS 63 

Now since this last is a constant quantity and independent 
of ///, it follows that the acceleration of the mean rates of 
increase of the product will be constantly the same during 
every two consecutive units of time, while the factors are 
increasing at a uniform rate. And since the increase of the 
variables is continuous and uniform, that of the product will 
also be continuous and according to a uniform law of some 
kind ; and since for every possible variation in the number 
and value of the units of time, and in the value of the rates 
dx and dy the acceleration of the rate of increase of the 
product is constant for successive periods, it must be so 
continuously, and equal to twice the product of the rates of 
increase of the variable factors. 

Proposition IV. 

(12) To find the differential of the product of two independent 
variables. 

Let us suppose the two variables A and B to be increasing 
at any rate whatever, either uniform or variable, and inde- 
pendent of each other. Suppose also that when A has 
become equal to x, B will have become equal to y, and that 
dx and dy represent their respective rates of increase at that 
instant ; then they will represent the uniform increments, 
that would be made by A and B respectively, in the same unit 
of time, at these rates ; and these suppositive increments ■ 
are what we have to consider. Suppose again that one-half 
of each increment be made immediately before A and B 
become equal to x and y, and the other half afterwards. In 
the first case the product of A and B, or AB, at the begin- 
ning of the increment will be equal to 

(**— J Adx)(y— %dy) =xy— %ydx— 1 / 4xdy + }{dxdy 
and at the end of the unit of time it will be equal to 

{x + y 2 dx){y + y 2 dy)-=xy + Y 2 ydx + y 2 xdy+]^dxdy 
Subtracting the first product from the last we have 

ydx-\-xdy 



64 DIFFERENTIAL CALCULUS. 

which represents the difference between the two states of 
the rectangle AB, or the increment made by it, while the 
factors are passing from x— T / 2 dx and y— %dy to x-\-%dx 
and y + %dy ; that is, while the variables are receiving the 
uniform increments represented by dx and dy, their respec- 
tive rates of increase at the instant they are equal to x and 
y, the rectangle is receiving an increment represented by - 
ydx+xdy. We are now to show that this increment repre- 
sents the rate of increase of AB at the moment that dx and 
dy represent the rates of increase of A and B separately; 
namely, at the instant they become equal to x and y. 

This suppositive increment would not, of course, be made 
at a uniform rate, but as we have seen (lemma) at a uni- 
formly increasing rate. Hence when AB would become xy y 
and the variables had received half their suppositive incre- 
ments, the increment of AB would have received half the 
increase of its rate, which would then have become equal to its 
mean rate for that unit of time. But the mean rate is that 
by which the increment would be made in the same time if 
it were uniform, and if the increment were made at a uni- 
form rate it would measure the rate of increase of the rec- 
tangle. Now the actual increment (represented by ydx-\-xdy) 
of the rectangle, being made in the same time, as it would 
be if made uniformly at its mean rate, existing when A and 
B w r ere equal to x and y y is the true measure of that rate j that 
is, of the rate of increase of AB at that instant ; or of xy if 
we consider x and_y as the variables. Hence the differential 
of the product of two variables is equal to the sum of the products 
arising from the multiplication of each variable by the differen- 
cntial of the other. 

Note. — This proposition being the key to the whole subject of the differential cal- 
culus, should be carefully studied and well understood. The result of this proposition 
might have been surmised by considering that a product of two variables is subject to 
two independent causes which produce its rate of change. If .x has a certain rate of 
increase, that of the product will, from that cause be y times that rate ; and if y have a 



DIFFERENTIATION OF FUNCTIONS. 



65 



B B' 



certain rate of increase, that of the product will from that cause be x times that rate : 
and the total effect of both causes will be the sum of the partial effects arising from 
each cause independent of the other; that is, the entire rate of increase of xy y is y 
times that of x plus x times that of y. This, however, is not mathematical proof — 
it only makes the result probable. Yet the text books asstcme vvithout any better proof, 
that the total differential is equal to the sum of the partial differentials, which is equiv- 
alent to the proposition as I have stated it. The truth is, it cannot be proven math- 
ematically by the method of infinitesimals nor limits. 

This proposition may be illustrated geometrically thus : 
Let x be represented by the line AB (Fig. 1), and y by 
the line ACj then the product xy will be represented by 
the rectangle ABDC. Suppose , P D ' 

x and y to be each increasing in 
such a manner that when x has 
become equal to AB, and is 
then increasing at a rate, that, 
if continued, would produce the 
increment BB' in a unit of time, 
y will have become equal to 
AC and be increasing at a rate 
that would produce the increment CC in the same unit of 
time. Then BB' will be dx and CC ' will be dy, for they 
will be the true symbols of the rates of increase of x and y. 
Now the uniform increment, BB' , of x will produce the 
uniform increment B JD E B r of the rectangle, which will 
therefore represent its rate of increase arising from that of 
x. In like manner CC FD will represent the rate of in- 
crease arising from that of y. Hence the symbol of the 
entire rate of increase of the rectangle arising from the 
rates of increase of the sides, will be the sum of the two 
rectangles B B' E D and C CF D or ydx+xdy. 

It must be remembered that the increments DE and DF 
are not actual increments given to the sides BD and CD, but 
suppositive increments, or symbols, representing their rates 
of increase ; for in order that the suppositive increment of 
the rectangle may be uniform, the sides BD and CD must 
5 



66 DIFFERENTIAL CALCULUS. 

remain constant. Hence the two rectangles BB'ED and 
CC'FD are not actual increments of the rectangle ABDC, 
but suppositive increments, wn^.i are symbols showing what 
would be its increment if the rate were to remain constant 
from that point in the value of x and y for a unit of time, 
and are, therefore, the measure of that rate. 

Note. — The increment of the rectangle arising from the actual increments of x 
andjy would include the small rectangle DED'F or dx . dy ; and hence it would be too 
great to represent the required rate by so much ; and it is to get rid of this surplus that 
the advocates of the infinitessimal theory, who take the actual increment to represent 
the rate, reduce it to an infinitessimal, which they claim may be neglected with impu- 
nity. But we see the true reason for throwing it out of the expression for the rate is, 
not because of its insignificance, but because, being produced by the actual increments 
of x and y after they have passed beyond the values of AB and AC, it has no connec- 
tion with the rate at which the rectangle was increasing at the moment x and y were 
equal to those values. 

Proposition V. 

(13) To find the differential of the product of any number 
of variable factors. 

Let us first take the product of three variables as 

xyz 
If we represent the product xy by u we shall have 

x y z—uz 
and (Art. 7 and 12) 

d (xyz) =d(uz) = zdu + tidz 
Replacing u by its value we have 

d(xyz) = zd(xy) -\-xydz 
or since 

d(xy) = ydx + xdy 
we shall have 

d(xyz) = zydx + zxdy + xydz 

If we take the product of four variables as 

xyzu 
and make xy—r and zu=s we have 

xyzu = rs 
and 

d(xyzu)=d(rs) =rds -\-sdr 



DIFFERENTIATION OF FUNCTIONS. 6j 

Replacing r and s by their values we have 
d{xyz&)=xyd(zu) -\-zud{xy) 
substituting the values of d(zu) and d[xy) we have 
d(x\ >2u) = xyzdu + xyudz + uzxdy + uzydx 

These examples may be carried to any extent, but are 
enough to show the law which governs the differential of a 
product, which law may be thus stated : The rate of change 
in a product arising from the rate of any one factor is equal 
to the product of all the other factors multiplied by that 
rate ; and the total rate of change in the product is the sum 
of all the partial rates arising from the rates of the factors 
taken separately. In other words, the total effect arising 
from all the causes acting together is equal to the sum of 
the partial effects arising from each cause acting separately. 

Hence, the differential of the product of any number of vari- 
able factors is equal to the sum of the products arising from mul- 
tiplying the differential of each variable by the product of all the 
other variables. 

EXAMPLES. 

Ex. i. What is the differential of ayz ? — 

A 11s . aydz + azdy 
Ex. 2. What is the differential of 4 bxy ? — 

Am. 4 b(xdy -\-ydx) 
Ex. 3. What is the differential of ax+byz? — 

Ans. adx + bydz + bzdy 
What is the differential of xy—uz? — Ans. 4 

What is the differential of $ax— 2xy ? — Ans. 
What is the differential of 2ay-t~$u? — Am. 
What is the differential of ^abxyz ? — Ans. 
What is the differential of bcu — a 2 —zy ? — Ans. 
What is the differential of ^axy — b 2 +cu? — Ans. 
What is the differential of ja 2 — 2bu+cy ? — Ans. 



Ex. 


4- 


Ex. 


5- 


Ex. 


6. 


Ex. 


7- 


Ex. 


8. 


Ex. 


?• 


Ex. 


10. 



6& DIFFERENTIAL CALCULUS. 

Proposition VI. 
(14) To find the differential of a fraction. 

CASE I. 

Let the fraction be 

x 

n 

in which the variable is divided by a constant. Make 

X 

then 

x=ny 
and (Art. to) 

dx=ndy 

hence 

that is, 

The differential of a fraction having a variable numerator 
and a constant denominator is equal to the differential of the 
numerator divided by the denominator . 

CASE 2. 

Let the fraction be 

n 
x 

in which the denominator is variable and the numerator 
constant. Make 



then 








n 

x=y 






and (Art. 


12 


and 


Art. 9 ) 


xy = n 






hence 






d(xy)=y 
dy- 


dx-\-xdy- 

_ _ ydx 
x 


-dn- 


-0 



DIFFERENTIATION OF FUNCTIONS. 69 

Replacing y by its value we have 

,n \ ___ x ax __ ndx 
\ x J X X 2 

that is, 

The differential of a constant divided by a variable is equal to 
minus the numerator into the differential of the denominator, 
divided by the square of the denominator. 

case 3. 



Let the fraction be 


X 


in which both terms are variable. Make 

X 


then 




— —U 

y 


and 




x=uy 


hence 


dx~- 


— d{uy) =ydu + udy 
du- dx ~ udy 


Replacing 


u by its value we have 




*?)= 


v J _ydx — xdy 




y y z 



that is, 

The differential of a fraction of which both terms are vari- 
able, is equal to the differential of the numerator multiplied by 
the denominator, minus the differential of the denomiiiator mul- 
tiplied by the numerator, and this difference divided by the square 
of the denominator . 

If the variables in any of these cases should be functions 
of other variables, we can represent them by single letters 
and replace them in the formula by their values. 

Thus, suppose the given fraction to be 



7° 



DIFFERENTIAL CALCULUS. 



make 
then 



u — x 
vy 

u—x=s and vy—r 

rds—sdr 



X~)=^> 



Replacing the values of s and r we have 

/ u — x \ vyd( u —x) — { u — x)d( vy) 

which being expanded becomes 



9 9 



<#£) 



vydu — vydx — uvdy — nydv + vxdy + xydv 



vy 



v 2 y 2 



EXAMPLES. 



Ex. i. 

Ex. 2. 

Ex. 3 

Ex. 4 
Ex. 5 
Ex. 6 
Ex* 7 
Ex. 8 
^r. 9 
.£.#. io 
i£#. ii 
Ex. 12 



What is the differential of ax+— ? 



Ans. #<r/,r + 



* 



What is the differential of cxy-\ — ? 

Ans. cxdy+cydx- 
What is the differential of ~~ j? Ans. 



adu 



What is the differential of 



a— b ■ 



Ans. 



%x 



What is the differential of 5^+^? Ans. 
W T hat is the differential of 3(#— #)— |^? Ans. 
What is the differential of 2ax-\-x (y—c) ? Ans. 
What is the differential of ^(by — x)(ti—c) ? Ans. 
What is the differential of ^- y x-^ 1 ? Ans. 
What is the differential of {a— v) (xy — z)? Ans. 
What is the differential of 6ab— $xy {jc—c) ? Ans. 
What is the differential of [z— y){a— x) + -~ ? Ans. 



DIFFERENTIATION OF FUNCTIONS. 7 1 

Proposition VII. 

(15) To find the differential of any power of a variable. 

We have seen (Art. 13) that the differential of the product 
of any number of variables is equal to the sum of the pro- 
ducts arising from multiplying the differential of each vari- 
able by the product of the rest. If there are n variable fac- 
tors in the given product, there will be n products in the 
differential, and the coefficient of the differential of each 
variable will be the product of (n—i) variables. If now all 
the variables become equal, we may represent each one by 
x, and the product will become x n , while each of the pro- 
ducts composing the differential will become x n ~ 1 dx; and 
as there are n of these products, the sum of them will be 
nx n ~ 1 dx, hence we have 

d (x n ) =nx n ~ 1 dx 

Hence, the differential of a variable raised to a power is equac 
to the variable raised to the same power less one, and multiplied 
by the exponent of the power and the differential of the variable 

If, for example, we take 

d (xyzu) =xyzdit -\-xyudz-\-xzudy -\-yzudx 
and suppose all the variables to become equal, we shall have 

dx 41 =4x 3 dx 
If we represents by y we shall have 

dy=nx n ~ 1 dx 
and 

dy_ n-l 
dx~ nx 

which is the differential coefficient of that function of x. 

(16) The rule here given, for finding the differential of 
the power of a variable, holds good for all values of n y 
whether integral or fractional, positive or negative. 

Case 1. Let n be negative and represented by —m, then 

1 



. Y> Tb ^33 a/* — TYh 3^33 



% 



m 



72 DIFFERENTIAL CALCULUS. 

and (Art. 14) 



\x m ) ' 



d(x m ) _ _mx m ~ 1 dx 

Dividing both terms of this fraction by x™- 1 we have 
mdx 

. — — <n; ir~~'W'~i-dx 



in accordance with the rule. 

Case 2. Let # be equal to — then 

x n — x m 

Representing x m by ^ we have 

.#— jy w 
and 

dx=my m ~~ 1 dy 
or 

*/# 

•^ 77iy m ~l 
Replacing jy by its value we have 

I 1) dx dx 1 i-i , 

m \x m ) mx 1 "^ 

r 

Case 3. Let n be equal to — and represent x m by jy and 

we have 

r_ 
y—x m or ^ =.%: r 
and 

my m ~ x dy = rx r ~ ^dx 
whence 

rx r ~ 1 dx 
# = my m-l 

Replacing y by its value, we have 

/ r_\ x r ^ 1 dx r ZL_i 

d\x™) = T - X , r , m -l ^-x^-m^-^dx^-x™ dx 



\x"\ 

hence the rule is true in all cases 



DIFFERENTIATION OF FUNCTIONS. 73 



Proposition VIIL 

(17) To find the differ e?itial of any root of a variable. 
Let us take the function 7 yTx and make it equal to y, 
then 

y n =x 
and 

my m - 1 dy=dx 
whence 

dx 
d y~ my m-l 

Replacing y by its value we have 

dx 



djx 



i\/x m, 

v m^/xr/i-l 

That is, the differential of any root of a variable is equal to 
the differential of the variable divided by the index of the root 
into the same root of the variable raised to a power denoted by 
the index of the root less one. 

Hence 

/ —_dx 

v 2\/ x 

These results could of course be obtained under the pre- 
ceding rule, by giving to the variable a fractional exponent. 
But this is not always convenient. 

(18) In all the cases which have been explained under 
these eight propositions, the single letters that have been 
used may represent functions of other variables, in which 
case the operation must be continued by substituting the 
functions for the letters representing them and performing 
upon them the operations indicated, which can be done by 
the rules already given ; for the terms of all algebraic func- 
tions must ultimately take some one of the forms that we 
have discussed. 



74 DIFFERENTIAL CALCULUS. 

Thus, if we have the function 
x 2 — *sj~y 
a—x 
we may represent the numerator by u and the denominator 
by v we shall then have 

X 2 — \/ y U 

a—x v 

and 

x 2 — A^fy vdu — udv 

(I 9 

a—x v* 

Replacing u and v by their values we have 

x 2 — Aj~y (a—x)d(x 2 — /y/"3/) — (x 2 — A x /~^)d(a—x) 

a—x (a—x) 2 

and performing -the operations indicated we have 

dy \ 

c 2 — <\/y (a—x)(2xa 7 x— 2 — J + (x 2 — ^/ y)dx 



x" 



a—x (a — x) 2 

which completes the differentiation. 

Proposition IX. 

(19) To find the differential of a function with respect to 
the independent variable when it enters the given function 
by means of another function of itself. 

Let there be a function of y in which y represents a func- 
tion of x. The proposition is to find the differential coeffi- 
cient of the given function with respect to x. Represent- 
ing the given function by u we have 

u=jF(y) and y=I?(x) 

In order to find the relation between du and dx, it might 
be supposed necessary to eliminate y from the equations and 
obtain one directly between u and x, to which the ordinary 
process of differentiation could be applied. But this is not 



DIFFERENTIATION OF FUNCTIONS. 75 

necessary; for if we differentiate these equations as they 
are we have 

dw=-F\y)dy and dy = E\x)dx 
in which F\y) and E\x) represent the differential coeffi- 
cient of // with respect to y and of y with respect to x. 

From the first of these we obtain 

, du 

and placing this value of dy equal to the other we have 
J^=F'(x)dx or %=F\y) . F\x) 

That is, the differential coefficient of u with respect to x 
is equal to that of u with respect toy multiplied by that of y 
with respect to x. Hence 

When the variable of a given function represents the 
function of an independent variable, then the differ e?itial 
coefficie?it with respect to the indepejident variable is equal to the 
product of the differential coefficient of the function with respect 
to the given variable, multiplied by the differential coefficient of 
the given variable with respect to the i?idependent variable. 

Thus if we have the function y 2 —ay in which y = 2a—x 2 , 
representing the given function by u, we have 



du j dy 

Zy = 2y - a and s = 



whence 

dx 



du o r 

-^ x — 2ax — 4xy : =4x 6 — Gax 



EXAMPLES. 



Find the differentials of the following functions in which 
the variables are independent. 

-Ex. 1. a 2 x 2 -\-z Ans. 2a 2 xdx-\-az 

Ex. 2. bx 2 — y 3 -\-a Ans. 2bxdx — 3^ 2 ^ 

Ex. 3. ax 2 —bx 3 -\-x Ans. 

Ex. 4. (c-\~d)(y 2 — x 2 ) Ans. 



76 DIFFERENTIAL CALCULUS. 

Ex. 5. 5^ 5 — 2<zy— b % Ans. 

Ex. 6 . x n — x 3 + \b A ns . 

Ex. 7. ax 3 — ifix Ans. 

Ex. 8. (.t 2 +^)(.t— <?) Ans. 

Ex. 9. # 2 jy a — s 8 Ans. 

Ex. 10. ^t 2 (^ 3 +^) ^//j-. 

a 

Ex. 11. 7 g ^^. 

isx 12. V# 2 — # 3 ^4//j\ 

Ex. 13. V2^T+,T 2 ^/w. 

i£r. 14. — . Ans. 
Vi— x 2 

Ex. 15. " ^>w. 

jr-J-V 1 — .t 3 

i^. 16. (# + a/*) 3 -^ ; ^- 



<7~ — x» 
Ex ' 17 ° a±+a*x 2 +x± 



Ans. 



EX ' l8 - ! + ^ AllS - 

I +* 3 

i^r. 10. ' t~ Ans. 

y i—x 4 

Ex. 20. ' _lf =? ^f/^. 

Ex. 21. (/z+A.%^)™ ^/^. 

Ex. 22. Vax' 6 Ans. 

1 JL 
./£#. 23. ^ 3 ^ -3 y^;/j-. 

3 

Ex. 24. ;;/^" 1 +(^ 3 j)^ ^;/j*. 

JL 
jfi-tf. 25. ^ 2 j 3 — Ans. 

JL 

-Zfo. 26. #+&\r— 3J 4 Ans. 

Ex. 27. cT -3 +_y _t -f-s 3 ^/^'. 

2 

£#. 28. {ax— y) 5 Ans. 



DIFFERENTIATION OF FUNCTIONS. 77 

Ex. 29. (i—c)(x—y)% Ans. 

Ex. 30. V x 2 +a *J~x Ans. 

Ex. 31 A person is walking towards the foot of a tower, 

on a horizontal plain, at the rate of 5 miles an hour; at what 

rate is he approaching the top, which is 60 feet high, when 

he is 80 feet from the bottom ? 

Let the height of the tower be a, the distance of the per- 
son from the foot of it be x, and the distance from the top 
be_y; then 

j/2 — cfi -\-x % 



and 
whence 



ycly—xdx 
, xdx 80/.'. x 5 m. 

#=-r= — m~ =4». 



Ans. 4 miles per hour. 

Ex. 32. Two ships start from the same point and sail, 
one north at the rate of 6 miles per hour, and the other east 
at the rate of 8 miles per hour ; at what rate are the ships 
leaving each other at the end of two hours ? 

Ans. 10 miles per hour. 

Ex. 33. Two vessels sail directly south from two points 
on the equator 40 miles apart, one sails at the rate of 5 
miles per hour, and the other at the rate of 10 miles per 
hour ; how far will they be apart at the end of 6 hours, and 
at what rate will they be separating from each other suppos- 
ing the meridians to be parallel? Ans. 3. miles per hour. 

Ex. 34.. A ship sails directly south at the rate of 10 
miles per hour, and another ship sailing due west crosses her 
track two hours after she has passed the point of crossing, at 
the rate of 8 miles per hour; at what rate are they leaving 
each other one hour afterwards ? 

Ans. 11.74 miles per hour nearly, 

Ex. 35. The vessels sailing as in the last case, how will 



78 DIFFERENTIAL CALCULUS. 

the distance between them be changing, and at what rate, 
one hour before the second crosses the track of the first ? 

At that time the first vessel will be 10 miles from the point 
of crossing, and the second 8 miles. Calling the distance 
of the first x, and the second y, the distance between them 
is V x 2 +y' z which we will call z/, then 

u—\/ x' z +y z 
and 

xdx + ydy _ 100 — 64 



du- 



V x z +y' z V 100 + 64 
We make 64 in the numerator negative because y is a posi- 
tive decreasing function, and its differential is therefore 
intrinsically negative. 

From the above equation we find that the vessels are sep- 
arating at the rate of 1.56 miles per hour nearly. 

Ex. 36. The height of an equilateral triangle is 24 inches, 
and is increasing at the rate of two inches per day ; how 
fast is the area of the triangle increasing ? 

Ans. Z 2 ^/ 3 square inches. 

Ex. 37. The diameter of a cylinder is 2 feet, and is in- 
creasing at the rate of 1 inch per day, while the height is 
4 feet and decreasing at the rate of 2 inches per day ; how 
is the volume changing? and how the convex surface ? 

Ans. The volume is increasing at the rate of 288 ~ cubic 
inches per day. The area of the convex surface is not 
changing # 



SECTION III. 



SUCCESSIVE DIFFERENTIALS. 

(20) In considering the differential of a function hitherto, 
we have regarded it as immaterial whether that of the inde- 
pendent variable was itself variable or uniform. In consid- 
ering the rate of change in the differential of the function, 
it will be most convenient to consider that of the indepen- 
dent variable as constant } and this we have a right to do, 
since, the variable being independent, its rate is always 
assignable. 

If we have the product of two or more independent vari- 
ables, whether they are alike as x 3 , or different as x.y.z, the 
value of the rate of change depends not only on that of each 
independent variable, but also on the absolute value of all 
the variables. Thus the value of d(x 3 ), or $x 2 dx, depends 
not only on that of dx, but also on the absolute value of x 2 , 
and is greater or less as x 2 is greater or less. So that $x 2 dx, 
which is the rate of change of x 3 , has its own rate of change 
or differential. 

To find this second differential we must treat the first as 
an original function ; and as x is supposed to change uni- 
formly, dx will be regarded as a constant. Now the func- 
tion $x 2 dx may take the form sdx.x 2 , and 

d{ $dx. x 2 )= ^dx.d(x 2 ) = 7,dx. ixdx =6xdx 2 
This is called the second differential of x 3 , being the differ- 

79 



So DIFFERENTIAL CALCULUS. 

ential of the differential. This order is indicated by plac- 
ing the figure 2 as a sort of exponent to the letter d, thus 
d' z (x 3 ) is the symbol for the second differential of „r 3 , and 
hence 

d^{x s ) = 6xdx 8 

If the function is at all complicated, and, especially, if 
we desire to indicate a differential coefficient, it is much more 
convenient to represent it by a single letter ; in which case 
the letter itself is, for the sake of brevity, called the func- 
tion. 

So that if we represent x z by u we shall have 

u=x z and d 2 u = 6xdx 2 (1) 

Since the second member of this last equation still con- 
tains the letter x, it will still have a rate of change which 
may be found by considering 6dx z as the constant coefficient 
of x, and differentiating we have (Art. 10) 

d 3 u = 6dx z ,dx = 6dx 3 (2) 

The expressions dx % and dx z are to be understood as 
indicating, not the differentials of x % and x 3 , but the square 
and cube of dx. - 

The figure placed as an exponent to the letter vindicates 
the order of the differential ; and the differential of any order 
above the first is the rate of change in the differential of the 
previous order. The differentiation of the differential can 
only take place while the latter represents what is still a 
function of the independent variable. The differential of 
an independent variable, being, as we have stated, supposed 
to be constant can have no differential. 

If we divide equations (1) and (2) respectively by dx 2 
and dx 3 , we have 

d 2 u d 3 u 



—6x and ~t~t—6 

dx* dx 6 

in which the second members are the second and third dif 
ferential coefficient of the function x s . 



SUCCESSIVE DIFFERENTIALS. 



8r 



(21) To illustrate the principle of successive differentia- 
tion, let us suppose A B D C F E G (Fig. 2) to represent a 
cube, of which the side A B is an increasing variable repre- 
sented by x. Let the suppositive increments Dd, Dd r and Dd" 
represent the three equal rates of increase of the three x's 
whose product is the given cube ; we are to find what will be 
the corresponding rate of increase of the cube itself. That 
is, Dd being equal to dx, what is the value of d(x 3 ). 

At the moment the sides of the cube become equal to x, 
the cube tends to expand by the movement of the three faces 
DA, DF and DG outward in the directions Dd, Dd' and 
Dd" , each face continuing parallel to itself, and thus increas- 
ing the cube. But in order that these increments may rep- 
resent the i'ate at which the cube was increasing when they 
began, they must be made uniformly at the same rate. 
Hence the areas of the faces must remain constant, and they 
must move at the 
same rate as the in- 
crements Dd, Dd f 
and Dd" are des- 
cribed ; the move- 
ment being con- 
trolled by those in- 
crements. Thus the 
three solids Da, Df 
and Dg, generated 
by the flowing out 
of the surfaces DA, 
DF and DG, with 
an unchanging area, 
at the same rate as 
when their sides be- 
came equal to x, 
form* the increment 



Fig. 2. 




52 DIFFERENTIAL CALCULUS. 

that would take place in the cube in a unit of time, at the rate 
at which it was increasing when its side, or edge, became 
equal to x ; and hence they form the true symbolic incre- 
ment which represents the rate of increase or differential of 
x s . Now each of the solids thus formed is equal to x 2 dx, 
and hence 

d(x 3 )=sx 2 dx 

But if the cube, when its edge is equal to x, is in a state 
of increase, the faces DA, DF and DG have other tenden- 
cies besides that of flowing directly outward. They also tend 
to expand in the direction of their own sides, and this ten- 
dency is quite distinct from the other. Let us examine and 
measure it. The tendency of the face DF to expand, aris- 
ing from that of the cube itself, would be by the flowing out 
of the sides DC and DE in the directions Dd and Dd \ and 
remaining parallel to themselves. The rate of increase of 
the face DF would be measured by the areas described by 
these flowing lines in a unit of time, at the same rate as 
when they became equal to x, their lengths being constant ; 
that is, by the areas Dc and De' . But the face DF 'of the 
cube is the base of the solid Df, and the tendency of the 
base to expand imparts a like tendency to the solid, and the 
rate at which the solid tends to expand is such that while 
the base would increase by the rectangles Dc and De', the 
solid would increase by the solids Dc and De" , which there- 
fore represent, symbolically, the rate of increase of Df. 
Now Dc and De" are each equal to xdx 2 , and hence the rate 
of increase of Df is equal to 2xdx 2 . But the differential 
of the cube is represented by three such solids as Df and 
the rate of increase of the whole, or the differential of the 
differential x s is 6xdx 2 . 

Again the solid Dc' or xdx 2 tends to increase in the direc- 
tion Dd' at such a rate as would generate the suppositive 
increment Dd'" , equal to dx 3 in the same unit of time and 



SUCCESSIVE DIFFERENTIALS. &$ 

in a uniform manner. Hence d(xdx 2 ) is equal to dx 3 , and, 
therefore, d(6xdx 2 ) is equal to 6dx 3 . 

(22) It must always be remembered that the solids rep- 
resented in the figure are not the actual increments of the 
cube, but the symbols which represent its rate of increase and 
the successive rates of that rate at the instant that x is equal 
to ABj that is, they are the increments that would take place 
in the cube, and in the increments themselves if made uni- 
formly. The law which governs the increase of the cube, 
contains within itself not only the rate at which the cube is, 
at any instant increasing, but also the law of change in that 
rate, and the law to which that law is subject ; and these 
symbols represent the development of that law which was 
actually operating at the instant the cube attained the value 
of AB 3 or x 3 , and before any farther increase had taken 
place. 

(23) We may learn from this demonstration the method 
by which the actual increment of a power may be developed. 
By dissecting the figure (Fig. 2) and noticing the parts of 
which the increased cube is composed, we find, first, the 
original cube or x 3 ; second, the three solids Da, Df and JDg 
or $x 2 dxj third, the three solids Dc ', De" and JDb which 

represent half the rate of increase of $x 2 dx or —— — ; and 

fourth, the solid or small cube JDd" ' , which represents one- 

6xdx 3 6dx 3 

third of the rate of increase of or ; and these 

2 2.3' 

make up the volume of the cube after being increased by 
the addition arising from the increment dx to the side AB. 
But these increments are suppositive, and are used merely 
as symbols to show the successive rates of increase, all of 
which exist in the function x 3 before any increment actually 
takes place. 

If we divide $x 2 dx by dx we shall have the first differen- 



84 DIFFERENTIAL CALCULUS. 

6xdx 2 
tial coefficient of x 3 . If we divide by dx 2 we shall 

have half the second differential coefficient of x 3 ; and if 

6dx 3 

we divide ■ by dx° we shall have one-sixth of the third 

2 . x 

differential coefficient of x 3 ; and these results, viz. : $x 2 y 

6x 6 

— and are the coefficients by which the successive 

22.3 J 

powers of dx must be multiplied in order to make up the 
parts composing the suppositive increment of x 3 . Now 
since these partial increments taken together with the orig- 
inal cube form also a complete cube, if we make dx a real 
increment and multiply its successive powers by these same 
coefficients, we shall have an actual increment of the cube, 
and the original x 3 will have become (x-\-dx) 3 . 

It must not be forgotten that these differential coefficients 
are true of the cube before the increment takes place, and 
when dx is equal to zero. 

For convenience we will designate the variable cube by 
u y and in order to mark the point where the differential is to 
be taken we represent its variable edge by (h+x) in which 
h represents the side AB, or that particular value of the 
variable where the differential is to be taken, while x will 
represent its variable increment and take the place of dx. 
Then the cube AE or ~ab? will be represented by u or 
(h+x) 3 at the time when the variable u equals h or x=o. 

This being premised we take the differential coefficients 
already found, namely, 3.T 2 , -^ and g- g, and substitute (h-\-x) 
for x (reducing ^ to zero at the same time) and x for dx in 
the other factors ; then 3X 2 becomes the first differential 
coefficient of u or (h+x) 3 , that is ~^- — 7,(h+x) 2 with x=o, 
or 3/2 2 ; y becomes ■ — l -^— with x=o; that is 3//, or half the 
second differential coefficient of u with x—oj and ^—3 



SUCCESSIVE DIFFERENTIALS. 85 

becomes one-sixth of the third differential coefficient of u or 



= 1. 



6,/v 3 ' 

Hence, indicating by a vinculum that x has been made 

equal to zero, we have for the three coefficients $x 2 , -% and 
g— g, which were true of the cube before any increment was 
made, 

tdu\ fd % u\ f d s u \ 

\ dx 1 V 2dx 2 1 ^ 2 • 3 • ^ 3 ' 

also true of the cube at the same time and the different parts 
of the cube increased will be represented as follows : 
The cube AEby {ti) or /i s . 

The three solids Da, Df and Dg by (jf-Xr or tJ&x. 

The three solids ZV, Ds n and Db by (^7^)^ 3 or 3/ix 2 . 

The solid Dd bv I T^)x 3 or ;c 3 

J ^2.3. ax 6 / ' 

and these make up the value of (//+.r) 3 , hence 

/^/// \ /// 2 u\ ( d z u \ 

u or {k+xY={u) + (^) * +(^)- 3 + (7T^)- 3 
or 

{zi+x) 2, =/i s + 3/1 2 x -}- 3/ix 2 -j-x' 6 

This illustrates to some extent the law which connects 
together the parts which go to make up the change in the 
function of a variable arising from that of the variable itself. 
A more complete and general demonstration of this law is 
contained in the following theorem. 

Maclaurin's Theorem. 

(24) A function of a single variable may often be ex- 
panded into a series by the following method. 



86 DIFFERENTIAL CALCULUS. 

Representing the function by u and the variable by x we 

shall have 

u^F^x) 

When this function can be developed, the only quantities 
that can appear in the development, besides the powers of 
x, will be constant terms and constant coefficients of those 
powers. Hence the developed function may be put into the 
following form : 

u=A+Bx+Cx* +Dx* +Ex± + etc. (i) 

in which A, B, C, D, E, etc., are independent of x. The 
problem is to find the value of this constant term A, and 
the values of the constant coefficients B, C, D, E, etc. For 
this purpose we differentiate equation (i) successively, and 
divide each result by dx y the successive differential coeffi- 
cients will then be 

du 

~i~=B + 2Cx+7 ) £>x 2 +4EX 3 + etc. (2) 

d*u 

~^ = 2C+2 . 3 . Dx+3 . 4 . Ex* + etc. (3) 

d 3 u 

-^- = 2 . 3 . £> + 2 .3.4. Ex+ etc. (4) 

Since x is an independent variable, these equations are 
true for all values of x, and, of course, when x=o. 

Reducing x to zero in equation (1), A becomes equal to 
the original function with x reduced to zero. We will rep- 
resent that state of the function by (u)j and also indicate by 
brackets around the differential coefficients that x=o in their 
values also. Then from equations (2), (3), (4), etc., we have 

and so on to the end of the development, if it can be com- 
pleted ; if not, then in an unlimited series. 

Substituting these values of A, B, C } D, etc., in equation 
(1) we have 



SUCCESSIVE DIFFERENTIALS. 87 

, . / du \ /d 2 u\x 2 ( d 3 u\ x 3 

«=(*)+te) * +(-^r)—+(-^r)^j+ etc - 

which is Maclaurin's theorem. 

EXAMPLES. 

Ex. 1. Expand (a+x) ?l into a series. 
Represent (a+x) n by u and we shall have 

u = (a+x) n =A+Bx + Cx 2 +£>x* + etc. (1) 

Differentiating we have 
du 

d 2 u 



dx 2 
d 3 u 



=n(n — 1) (a-\-x) n ~% 



=n{n— i){n— 2)(a-\-x) n ~Z 
4- —n(n—i)(n—2)(n—T > (a+x) n ^ 4: 



dx' 6 
d±u 



dx 

from which, when x=o, we have 
A =a n 
B — 7td rh ~ 1 

n(n—i) 
C = ; a n ~* 

2 

n(n—i)(n—2) 
D— -a n ~* 

2 - 3 

n(n — i)(n — 2){n— $)a n '~4 

E == 

2 • 3 • 4 

Substituting these values in equation (1) we have 

and so on ; the same result as by the binomial theorem. 

Ex. 2. Develop — x into a series. 
We have by differentiation 



88 DIFFERENTIAL CALCULUS. 



die 



dx (a+x) 2 

d 2 U 2 

dx 2 (a-j-x) s 

d z u 2 . 3 



dx' 6 (a+xY 

and so on. 

Making x— o in the values of u and the differential coef- 
ficients we have 

W~~a> \7x')~'~~a 2 > \~dx T )~~a^ \~dx^) = ~~^ 
Substituting these values in Maclaurin's formula we have 

~3 



v— — ; — —— — — o+ — T~"~ T+ etc. 



I I X X * X 

a+x a a 2 a 3 a ■ 

Ex. 3. Develop the function ^z^ into a series. 

^ttf. i+^+^ 2 +^ 3 +^ 4 + etc. 
,£!#. 4. Develop the function /y/# + x into a series. 

JL _JL i _ 3. j 3 _!. 

^/w. a 2 -\-\a 2x ~^^ 2 x 2 + 2 4 6 <? 2 ^c 3 — etc. 

At. 5. Develop / _■_ „\2 m to a series, ^tw. 

jSt. 6. Develop 3 ^/(«4j)2 into a series, ^;w. 
Ar. 7. Develop V# 3 +£r into a series. Ans. 

1 

At. 8. Develop ±/~YZL — mto a senes - Ans. 

Ar. 9. Develop {a 2 — x 2 ) 3 into a series. Ans. 

The formula of Maclaurin applies in general to all the 
functions of a single variable that are capable of successive 
differentiations. But there are cases in which the function 
or some of its differential coefficients become infinite when 
x~o j in such cases the formula will not apply. The func- 

tion, c+ax 2 is an example of this kind; for if we represent 
it by u we have 



SUCCESSIVE DIFFERENTIALS. 89 



and 



1 

//=,- + (7 X- 



au 1 a 

2X~ 



If in this we make x=o for the value of the coefficient 
jB, we have 

In general, any function of x in which x is not connected 
with a constant term under the same exponent, cannot be 
developed by this theorem ; for the differential coefficients 
will be such as to reduce to zero or infinity in every case, 
when x is made equal to zero. 

Taylor's Theorem. 

(25) The object of this theore?n is to obtain a formula for the 
development of a f miction of the sum or difference of hco vari- 
ables. 

The principle on which this theorem is based is the fol- 
lowing: The differential coefficient of a function of the sum 
or difference of two variables, will be the same whether the 
function is differentiated with respect to one variable alone, or 
to the other variable alone. Thus the differential coefficient 
of (x-\-v) u will be ^(.t + v) ?;_1 if we differentiate with res- 
pect to either variable alone, the other being considered as 
constant. 

A function of the sum or difference of two variables is one 
in which both are subject to the same conditions, so that the 
value .of their sum or difference might be expressed by a 
single variable without otherwise changing the form of the 
function ; and hence we may regard this sum or difference 
as itself a single variable. Now any rate of change in one 
of the two component parts (the other being regarded as 



90 DIFFERENTIAL CALCULUS. 

constant) will produce the same rate in the compound vari- 
able (so to speak) as it has itself; thus x+y will increase at 
the same rate as x if y be constant, and at the same rate as 
y if x be constant. So that changing from one to the other 
is merely changing the rate of the single variable that would 
represent the value of their sum or difference. But such 
change in the rate will not change the ratio which it bears 
to the corresponding rate of the function (Art. 6) ; that is it 
will not affect the differential coefficient. 

(26) To apply this principle let us take any function of 
the sum of two variables, as F(x-\-y), which we will repre- 
sent by u. If it can be developed into a series, the terms 
of the series may always be arranged according to the power 
of y; the coefficients being functions of x and the constants ; 
hence it may be made to take the following form : 

u=F&+y)=A+£y+Cy 2 +l?yZ+£y 4 <+ etc. (i) 

in which A, B, C, D, E, etc., are independent of y, but 
functions of x. 

If we differentiate equation (i) regarding y as constant, 
and divide by dx, we shall have 

du dA dB dC dD 

^ = ^ + ^ ;+ dxr-> ,2+ dx~^+ etc - 

If we regard x as constant, and divide by dy, we have 

du 

j-<=JB.+2Cy+3Z)y*+4Ey>+ etc. 

and since ^ is equal to -^ the second members of these 
equations are equal ; and since this equality exists for every 
value of y, and since the coefficients are independent of that 
value, the corresponding terms containing the same powers 
of y must be equal each to each ; hence 



SUCCESSIVE DIFFERENTIALS. 91 

dA 

dx~= B ^ 

dB 

^=* C (3) 

dC 

-T=iD (4) 

dD 

^=** (5) 

If now we make y =1 o, then B(x-r-y) becomes jF{x), which 
we will represent by z. Under this supposition equation (i) 
will become 

u (now become z) = A 
Substituting this value of A in equation (2) we have 

Substituting this value of B in equation (3) we have 
d% d*2 



whence 



dx d* 2 



d 2 Z 

c=- 



2dx 2 



similarly we have 

d s z d 3 z 



2dx 3 ° 2 . 3 . dx 3 
and 

dH _ d±z 

2.3. dx 1 ~ 4 ~" 2 . 3 . 4 . dx 4 " 
and so on. 

Substituting these values in equation (1) we have 

dz d 2 z d 3 z 

in which the first term is what the function becomes when 
y=o, and all the coefficients of the powers of y are derived 
from it on the same supposition. 



92 DIFFERENTIAL CALCULUS. 

This is the formula of Taylor. 

A function of x—y is developed by the same formula by 
changing y into —y, thus : 

/J? d z d 1 ? 

"=f(x-}>)=z-^y+^-iy 2 - 2 . 3 \j x3 y 3 - etc 

EXAMPLES. 

Ex. i. Let it be required to develop (x+y) n . 
Representing this function by u we have 
u=(x-\-y n ) and z=x n 
then by differentiation 

dz . d 2 z d*z 

Substituting these values in the formula we have 
u={x+y) n =x n +nx"-ly+ n ^ ) x n -Y + *% 1) £~V -y + 
etc., the same as by the binomial theorem. 
Ex. 2. Develop the function \/ x+y. 

JL ' JL _JL i _3_ _ -^ 3 _5. 

^/w. (x+y) 2 —x 2 +^x 2 y—^x- 2 y 2 +^^x 2 y s — etc. 
Av. 3. Develop \/ x+y 

JL _3. r 2 -£ 2 5 — & 

^//^. # 3 +i# 3 J ; — 37^ ■* 3 7 2 + 3 ~ 6 ' 9 # 3 _y 3 — etc. 

i£#. 4. Develop the function (#■— ^) ?l . yto. 

i£x. 5. Develop the function (x—y) 2 . Ans. 
Ex. 6. Develop the function ;— . ^//j. 

-2. 

i£x. 7. Develop the function of #(x— _j) 3 . Ans 
(27) Although a function of the sum or difference of two 
variables can generally be developed by this formula, yet 
there are cases in which the coefficients (which are functions 
of one of the variables) may, by giving certain values to the 
variable they contain, become infinite. In such cases t v e 
formula cannot be applied; for in general such values for 



SUCCESSIVE DIFFERENTIALS. 93 

that variable, would not reduce the function itself to infin- 
ity, although it would have that effect on its development. 
Thus in the function 

u=a+ {b—x+y) 2 
we have 

i_ dz i d 2 z 1 



z—a-\-{b— x) 2 , 



UX 2(b~xY ^ 4 {b~x) 2 



and so on. 

If now we make.r=£, all the coefficients will become infi- 
nite, and we should have 

_i 
u=a + (fr— x+y) 2 =a+ co 

by the formula instead of having as we ought 

jl 

2i—a-\-y 2 ; 

which cannot be, for the value of y is not dependent on 
that of x, and hence u is not necessarily infinite when 
x—b; but for all other values of x the formula will give the 
true development of the function. 

And herein is the difference between the formulas of Tay- 
lor and Maclaurin ; when that of the former fails it is for 
only one value of the variable ; while that of the latter when 
it fails at all, fails for every value of it. 

Note. — In fact, the theorems of both Taylor and Maclaurin are founded on the 
principle illustrated in Art. 21. The real object of both is to find from the rate of 
change of a function what will be its new state arising from a given change in the value 
of the variable. 

The general method of doing this is to find the successive differentials of the func- 
tion in its first state, and then to multiply the successive differential coefficients by the 
successive powers (properly divided) of the actual change in the variable, this will give 
the actual successive partial changes in the function which together make up the 
entire change, and thus develop the function in the new state. 

For this purpose the variable must have two points of value ; one where the func- 
tion is to be differentiated, and the other, the new value produced by the change ; and 
to this end the variable, in algebraic functions, is made to consist of two parts, either 
by making it a binomial or something that may be reduced to that form. 

In Maclaurin's theorem the variable consists of a constant and a variable combined 
together, so that their united value is a variable one, and the constant part is simply 
cne point in that variable value. This is the point at which the differentiation of the 



94 DIFFERENTIAL CALCULUS. 

function is made ; but as a constant cannot be differentiated, the variable is attached 
to it long enough for that purpose and then made zero. In Taylor's theorem the varia- 
ble is the sum or difference of two others, and the poirrt of differentiation is when the 
variable has reached the value of one of its variable parts. This being a variable, the 
function can be differentiated directly, and the other variable may be made zero be/ere 
the operation. Hence the theorems of Maclaurin's and Taylor are alike in this : both 
■ have a compound variable having two points of value, both are differentiated at the 
same point, and the successive differential coefficients, which are precisely alike in form 
and value, are multiplied by the successive powers of the change in value. The only 
difference is that in one the differentiation at the required point is made indirectly , and 
the variable change made zero afterwards ; while in the other the differential is made 
directly, the variable change being made zero beforehand. Hence a function of a 
binomial variable may be expanded by either method. By Taylor's, considering both 
terms variable and reducing one to zero before differentiation ; or, by Maclaurin's, by 
considering one term as constant and reducing the other to zero after differentiation. 
Thus in the case of the function \X~\~y) 71 ^ tne differential coefficient will be pre- 
cisely the same if we reduce y to zero and differentiate x by Taylor's method, or con- 
sider x as constant and reduce^ to zero after differentiation, by Maclaurin's method. 

In order that a binomial may represent a single variable, both terms must be subject 
to the same conditions, so that each term may be considered as a part of the same 
compound variable ; and the failing cases in Maclaurin's and Taylor's methods are 
simply those in which the binomial variable becomes a monomial, by giving the variable 

JL 
a certain value. Thus the case cited in Art. 24, C -f~ (IX , is not a true binomial vari- 
able, since the terms are not subjected to the same conditions. If we make it 

1 
\f-\-ClXy we have a true binomial variable, and the differential coefficient 

_1 

\\C~\~x\ will not reduce to infinity when x=o. Similar^ the case cited in Art. 

1_ 

27, namely, u' = -(l-\-{b — X-\-y)^ is one in which when x—h, the variable in the 

A 

function reduces to y, and the function itself to <2-J-y 2 , which does not contain a 

binomial variable of the required form. 

The same principle will apply to transcendental functions ; which, in order to be 
developed, must have two points of value in the compound variable — one for the dif- 
ferential and the other for the development. Thus Cl X k may be expanded by Mac- 
laurin's theorem, since it has two points of value, one at k, the point of differentiation 
where x=o, and the other the full value produced by x. 



SECTION IV, 



MAXIMA AND MINIMA. 

(28) We have seen that when a variable changes its value 
at a uniform rate, the value of its function will in general 
vary at a rate that is not uniform. It may increase at a 
diminishing rate, until at a certain point it ceases to increase 
and begins to diminish, in which case the turning point is 
the one of greatest value, and is called a maximum. Or it 
may decrease to a certain point and, having attained its min- 
imum, begin to increase. The problem is to find whether 
there is a maximum or minimum value for a function, while 
its variable is uniformly increasing, and if there is, to find 
the corresponding value of the variable and its function. 

(29) While a positive function is increasing as the varia- 
ble increases, its rate of change or differential will be posi- 
tive ; and negative when it decreases (Art. 3). Hence when 
a function is passing through a maximum or minimum value, 
the sign of the differential coefficient must change from 
minus to plus or from plus to minus — the former in case of 
a minimum, and the latter in case of a maximum. 

But such change can only take place while the differential 
coefficient is passing through zero or infinity. Our first 
inquiry then is whether there is any finite value of the vari- 
able that will reduce the first differential coefficient to either 



96 DIFFERENTIAL CALCULUS. 

of these values. For this purpose we solve the equation 
formed by placing the first differential coefficient equal to 
zero, and thus find the corresponding value of the variable. 
Here we have one of three results. 

First. There may be no real value for the variable. In 
this case there is neither maximum nor minimum. 

Second. There may be a real finite value for the variable 
that will reduce the differential coefficient to zero. In this 
case there ^\\\\ probably be a maximum or minimum. 

Third. There may be no finite value of the variable that 
will reduce the differential coefficient to zero, but at the same 
time there may be one that will reduce it to infinity. In this 
case we form the equation by placing the differential coeffi- 
cient equal to infinity, and the root that satisfies the equa- 
tion will indicate a probable maximum or minimum. 

In order to determine in the two latter cases whether there 
is a maximum or minimum value of the function, and if so 
which of the two it is, we may substitute in the function, in 
place of the variable a quantity a little less, and one a little 
greater than that derived from the equation. If the result 
in both cases is less than when the true value is substituted 
there is a maximum ; if greater, there is a minimum value 
of the function for the true value of the variable. 

We may also determine the same thing by substituting 
these approximate values in the differential coefficient, which 
the true value reduces to zero. If they cause the result to 
change the sign from plus to minus by substituting first the 
less, and then the greater quantity, there is a maximum, for 
the function is passing from an increasing to a decreasing 
state. If the change is from minus to plus, there is a min- 
imum, for the function is passing from a decreasing to an 
increasing state at that point. 



MAXIMA AND MINIMA. 



97 



EXAMPLE. 



Find the value of x which will render u a maximum or 
minimum in the equation, 

u=x 3 — 9.x 2 +24.T — 7 
Differentiating and placing the differential coefficient 
equal to zero we have 

d £=3x 2 — iSx + 24 = 3(x 2 — 6x + 8)=o 

from which we find 

cc=4 and x=2 
If we substitute in the function and in the different coeffi- 
cient 1, 2, 3, 4, 5, etc., successively, we shall have for 

du 





X- 


= 1 . . 


. w- 


= 9 • 


• dx 3 




X- 


— 2 . . 


. u- 


= 13 • 


du 

• dx~° 




X- 


=3 • • 


. W- 


=11 . 


du 

• dx~~ 1 




X- 


=4 • • 


. w- 


= 9 • 


du 




X'- 


=5 • • 


. u- 


= 13 - 


du 

* dx 3 




X 


=6 . . 


. u 


= 29 . 


du 

■ ■ ax = 2 4 


ndicating 


that 


for x- 


— 2 


the vs 


due of th< 



maximum, the differential coefficient passing from plus to 
minus ; and for x=4 the value of the function is a mini- 
mum — the differential coefficient passing from minus to plus. 

(30) It must be understood that by maximum and mini- 
mum is not meant the absolutely greatest or least value of 
the function, but the turning point, from an increase to a 
decrease, or vice versa. Hence there may be as many max- 
ima or minima of the function as there are values of the 
variable that will reduce the first differential coefficient to 
zero or infinity. 

It is also to be understood that in the discussion of this 
subject, when a function is stated to be an increasing one, it 
6 



98 DIFFERENTIAL CALCULUS. 

is meant that it is either positive and becoming greater, or 
negative and becoming less. If it is said to be decreasing, it 
is either positive and becoming less or negative and becom- 
ing greater. Thus if we take the function 

*=*• — 25 

and make x, successively, equal to 

1.2.3.4.5.6.7. 
the successive values of the function will be 
-24,-21,-16,-9, o, +11, + 24 

and it is said to be increasing throughout the whole change, 
although at first its numerical negative values decrease. 
This is also indicated by the sign of the differential coeffi- 
cient which is positive as long as x is positive. The terms 
" increasing " and " decreasing " then, in this case, refer 
merely to the direction in which the function is changing, no 
matter on what side of zero its value may be. 

(31) There is another method of ascertaining whether the 
first differential coefficient changes its sign on passing 
through zero or infinity, for this is the unfailing test of a 
maximum or minimum. Having found that value of the 
variable which reduces the first differential coefficient to 
zero, substitute that value in the second differential coeffi- 
cient, if it contain the variable, then 

First. If it reduces the second differential coefficient to 
a negative quantity, it indicates that when the first is at zero 
it must be a decreasing function, which can only be at that 
point by its passing from a positive to a negative state, and 
hence the function itself must be passing from a state of 
increase to a state of decrease, and hence is at a maximum. 

Second. If it reduces the second differential coefficient to 
a positive quantity, it indicates that the first when at zero is 
an increasing function, and must, therefore, be passing /n?//z 
a negative to a positive state, hence the function is passing from 



MAXIMA AND MINIMA. 99 

a decreasing to an increasing state, and is, therefore, at a 
minimum. 

Third. If it reduces the second differential coefficient to 
zero, we may resort to the third ; and if the same value of 
the variable reduces that to a read finite quantity, either pos- 
itive or negative, it shows that the second, at zero, is chang- 
ing its sign, and, therefore, the first is changing from an 
increasing to a diminishing function, or vice versa, and, 
therefore, does not at the zero point change its sign. Hence 
there is neither maximum nor minimum in the value cf the 
function. 

Fourth. If it reduces the third differential coefficient to 
zero we may resort to the fourth. If it reduces this to a real 
finite value, it indicates that at zero the third changes its 
sign, for it can only increase on both sides of zero by pass- 
ing from negative to positive, or diminish on both sides by 
passing from positive to negative. This will show that the 
second coefficient does not change its sign, for if it increases 
on one side of zero and decreases on the other, or vice versa, 
it can only approach the zero point until it touches it, and then 
must recede without changing its sign. This proves that the 
first coefficient does change its sign, for since the second does 
not change the first must be passing through from one side 
to the other. There will, therefore, be a maximum or mini- 
mum — the first if the fourth differential coefficient has a 
negative value, and the second if it is positive. 

We may continue thus and show that if the first differen- 
tial coefficient that is reduced to a real value, by substituting 
that value of the variable that reduces the first to zero, is of 
an even order and positive, there will be a minimum ; if it is 
negative, there will be a maximum ; and if it is of an odd 
order there will be neither maximum nor minimum. 

Fifth. If any value of the variable reduces the first dif- 
ferential coefficient to infinity, it will probably reduce all the 



IOO DIFFERENTIAL CALCULUS. 

succeeding ones to infinity, also. It will, therefore, be best 
in such a case to substitute values for the variable a little 
less and then a little greater than the one found. If the 
value of the first differential coefficient changes from plus to 
minus there is a maximum, and the second will be plus on 
both sides of infinity ; for the first must be an increasing pos- 
itive function in order to become positively infinite, and if 
negative on the other side must be a decreasing function, 
for it cannot be an increasing negative function on leaving 
infinity. Hence (Art. 3) the second must be positive in both 
cases. 

Sixth. If the first differential coefficient in the last case 
changes from minus to plus there will be a minimum, and 
the second coefficient will be minus on both sides of infinity. 
Thus we see that when any value of the variable reduces 
the first differential coefficient to zero, and is substituted 
in the second, a minus result indicates a maxi?num in the 
function, and a plus result a minimum. When any value 
reduces the first coefficient to infinity, a plus sign for the 
second indicates a maximum, and a minus sign a minimum. 

EXAMPLES. 

Ex. 1. In order to illustrate the first case in this article 
we take the function 

u — \6x — x 2 (1) 

from which we obtain by differentiation 

du 

dx~ =l6 ~ 2X (2 > 

d % u _ _ 

dx 2 " 2 KZ) 

We find that x=S will reduce the first differential coefficient 
to zero, while the sign of the second is minus. Hence at 
#=8 the first must be a decreasing function, and, therefore, 



MAXIMA AND MINIMA. 



IOI 



passing through zero from plus to minus, the function will, 
therefore be an increasing one to that point and then a 
diminishing one ; hence a maximum. 

If we substitute in the function and the first differential 
coefficient for x values a little less and a little greater than 
8, we have for 



x=S 



- du 

:6 3 • • cTx^ 2 



^ = 64 . 



du_ 
dx~ 



du 




If we represent the values of the function by the ordi- 
nates of the curve ABC (Fig. 3), the curve itself will cor- 
respond to the range or locus of values of the function, while 
the variable increases uniformly in passing from 7 to 9. 
From A to B the function increases, 
but at a decreasing rate, and hence 
the first differential coefficient is 
positive but decreasing until it 
reaches zero at B. The function then 
decreases at an increasing rate, and 
hence the first differential coefficient 7 $ $ 

must be negative and increasing. Fi s- 3. 

But when this coefficient, (or any variable) is positive and 
decreasing, or negative and increasing, its rate of change, i.e., 
the second differential of the function, must (Art. 3) be neg- 
ative throughout, which corresponds with the result found in 
equation (3). 

Ex. 2. To illustrate the second case we take 

u=x % — 16^ + 70 (1) 

from which 

du 

&=**-* 00 

dx* ~ 2 ^ 3) 



x=j ^=7 



102 DIFFERENTIAL CALCULUS. 

We infer from equation (3) that ~ x is an increasing func- 
tion for all values of x, and hence it is so when x=8, which 
reduces ^ to zero. From which we infer that -^ is passing 

from a negative to a positive state, and the function itself 
from a decrease to an increase. Hence a minimum. 

If we substitute 7, 8 and 9 successively for x in equations 
(i)-and (2), we have for 

du 

dx~~ 2 
r du 

z O -r — O 

dx 

du_ 

dx~ 
which corresponds with our deductions. 

If we let the ordinates of the curve ABC (Fig. 4) rep- 
resent the values of u, we see that from m ^ 
A to B the value of u diminishes, as is 
shown by the sign of c ~, and at a dimin- 
ishing rate as is shown by the positive 

sign of — J-J- (Art. 3). From B to C u Fig. 4 . 

increases, as is shown by the positive sign of ^, and at an 

d' 2 it 
increasing rate, as is shown by the sign of —~, , which is 

still plus. Hence the shape of the curve. 
Ex. 3. To illustrate the third case we make 

u— 9 ~\- 2 (x— 3) 3 (1) 



whence 



^=6(x- 3 Y (2) 

d 2 u 



dx 2 



= I2(*— 3) (3) 

Here we find #=3 reduces ^ to zero, and hence if there 
is a maximum or minimum it will be for that value of >#, 



MAXIMA AND MINIMA. 



103 



d 2 U 



But it also reduces -~rj- to zero also, hence we resort to the 



value of 



d ^ u 



that when 



du • 



dx 3 ' 
d 2 u 



which we find to be 12. We infer from this 



dx 2 



=0 it is passing from negative to positive, 



hence -£■ is passing from a decreasing to an increasing func- 
tion at the zero point, and, therefore, does not pass through 
it. It is, therefore, all the time positive, and the function is 
at all times an increasing one, so that there is neither max- 
imum nor minimum. We may. also, learn the same thing 

from inspection, for since the value of ~ n is a square it must 

always be positive. 

If we substitute in the given function and in the values 

du d 2 u 

of -7— and ~t~y the numbers 2, 3 and 4 successively for x, 

we have for 



x —3 u—9 



die 
dx 

du 
dx 



d 2 u 



dx 2 
d 2 u 



= — 12 



x=4 



du 

u=-\\ ~r~— 6 

dx 



dx 2 
d 2 u 



dx 2 



: I2 



If we let the ordinates of the curve ABC (Fig. 5) repre- 
sent the values of u, we see that from 2 to 3 the function 
increases, as is shown by the positive 
value of 00 but at a decreasing rate, 

as is indicated by the negative sign of 

d 2 u 

-^2" (Art. 3), be'tween those points 

or while x is less than 3. From 3 to 
4 the function is still increasing, as is 




104 DIFFERENTIAL CALCULUS. 

shown by the positive sign of ^, and at an increasing rate, 

d 2 u 
as is shown by the positive sign of -j~y when the value of 

x is greater than 3. Hence at B where the value of -^ is 
zero, the function having increased at a decreasing rate up 
to that point, ceases for an inappreciable moment, and 
begins again to increase at an increasing rate. 
Ex. 4. To illustrate the fourth case take 

u= 5 +(x- 7 y (1) 

du 

—= 4 ( x -jy ( 2 ) 

d 2 u 

-^- = I2{x- 7 )2 (3) 

d ** u 

•^s- = 24(*-7) (4) 

dx 4 



whence 



■= 2 4 (5) 



Here we see that the fourth differential coefficient is the 
first that has a real finite value for x=j, which reduces all 
the preceding ones to zero. Hence, according to our rule, 
there should be a minimum value for the function at that 
point. In fact, the sign of this coefficient shows that the 
third changes at zero from minus to plus ; and this shows 
that the second does not change its sign at zero, but after 
being a decreasing function to that point, becomes an in- 
creasing one, and is, therefore, positive both before and after. 
And this again shows that the first does change its sign at 
zero, since it is an increasing function on both sides of zero, 
which can only be by passing from a negative to a positive 
value. Hence the function will decrease until x : =j, after 
which it will increase, showing a minimum at that point. 

If we substitute in the given function and in the differen- 



MAXIMA AND MINIMA. 105 

tial coefficients the numbers 5, 6, 7, 8, 9 successively for x 
we shall have for 



*=5 


7/ = 1 1 


du 

dx $ 


^/ 3 & 
dx 2. 


-48 


d 3 u 
dx 3 


— 48 


x = 6 


z/— 6 


u 


4 


a 


= 12 


a 


= — 24 


X = J 


?/=: 5 


" = 




a 


= O 


a 


= O 


x = S 


u= 6 


" - 4 




a 


= 12 


a 


= 24 


x — 9 


^ = 11 


" = 3* 




a 


-48 


a 


= 48 



u ' -^^+x^ • ~+-j-$ • : — : + etc. 



which illustrates the conclusions we have drawn. 

The general proposition enunciated in the fourth case may 
be demonstrated analytically as follows. Let us suppose 

u=J?(x) 
and let the variable x be first increased and then diminished 
by another variable h ; and let these new states of the 
function be represented by 11 and u\ then we have 
u r —F{x+h) 
u!'=fXx-Ji) 
Developing these by Taylor's theorem we have, after sub- 
tracting the original function #, 

du d 2 u h 2 d 3 u 
"dx dx 2 ' 2 dx 3 '2.3 

du d 2 u h 2 d 3 u h 3 

u —u — —-j-/z-{- , o . — — ? o . + etc. 

dx dx* 2 dx 6 2 . 3 

Since the powers of h increase in each successive term of 
this development, we may reduce the value of h to such an 
extent that the value of any one term shall be greater than 
that of all the succeeding terms added together. Such in 
fact will be the case if h is less than one-half in the series 
h, A 2 , /z 3 , etc. 

Let us suppose h to be so reduced, then if u is a maxi- 
mum, it must be greater than u f or u ', and the second mem- 
bers of both these equations must be negative ; if it is a 
minimum it must be less than id or u\ and in this case the 
second members of both equations must be positive. 



d 2 u \ 
minimum. If the second term (or , 2 J become zero, there 



106 DIFFERENTIAL CALCULUS. 

Hence, in case of a maximum or minimum, the second mem- 
bers of both equations must have the same sign, and the 
first term of each (which controls the value of all the rest) 
having contrary signs must reduce to zero ; that is, ^ must 
be zero, since h is not. This then is a necessary condition 
to a maximum or minimum. If there is a real value for the 

. d 2 u 

second term in each equation, its sign (or that of • 2 - 

since h 2 is positive), will now control that of the whole sec- 
ond member, and will determine whether u is a maximum or 

d*u \ 

dx 2 

can be no maximum nor minimum unless the third term (or 

d 3 u\ 
j 3 ) which is now the controlling term, is also zero, since 

it has contrary signs in the two equations. We see then 
that the conditions of a maximum or mimimum are : first, 
that the first differential coefficient should become zero ; 
and, second, that the first succeeding differential coefficient 
that has a real value should be one of an even order, since the 
even terms have the same sign in both equations. If that 
is negative, the whole of the second member of the equa- 
tion is negative, and there is a maximum ; if it is positive, 
there is a minimum. Which agrees with the rule already 
found. 

Ex. 5. To illustrate the fifth case we take 



whence 



^io- 


— {x- 


2 
-3) T 


du 




■2 


dx ~ 


30- 


-3)* 


d 2 u 




2 


dx 2 


4 
9 (^- 3 ) 3 



(0 
(2) 
(3) 



MAXIMA AND MINIMA. 



ioy 



Here we find that x—^ will reduce c -£ to infinity. Refer- 

d 2 u 
ring to the value 01 , z we find that it reduces that also to 

infinity, but we see by inspection that any other value for 

d 2 it 
x, whether greater or less than 3, will make that of , 2 

positive. We see also that ~ will be positive when x is less 
than 3, and negative when it is greater. From all this we 
infer that the function is an increasing one before x=3, and 
a decreasing one afterwards. Hence there is a maximum 
at that point. 

If we substitute for x, in equations (1), (2) and (3), the 
numbers 2.3.4 successively, we have for 

du d 2 u 

x-2 u = 9 -7- = ° 



X- 



dx 


— 3 


dx 2 


du 

dx 


= 00 


d 2 u_ 
dx 2 


du 
dx 


2 

— 3 


d 2 it 
~dx~ 2 = 



x= 4 u= 9 -y- = 



If we let the ordinates of the curve ABC represent 
the successive values of u (Fig. 6) we 
see that from 2 to 3 the function in- 
creases, as is shown by the positive 
value of ^, and at an increasing rate 

as is shown by the positive value of 

d 2 it 
, 2 - From 3 to 4 the negative sign 

of ^- indicates a decrease of the function, while the positive 

sign of ~r~2~ (which has not changed) shows that this decrease 

is at a decreasing rate. Hence the form of the curve. 
Ex. 6. To illustrate the sixth case we take 




io8 



DIFFERENTIAL CALCULUS. 



whence 



, = ( 3 ^_ 9 )3 



du 



(IX f \ -Q 

d 2 u 2 



dx* 



( 3 ^- 9 ) 3 



(i) 

w 

(3) 



du 



In this case, as before, we find that x=$ reduces ^- and 



d 2 u 

~dx~*~ 



to infinity. We see also by inspection that ~r~ changes 

from minus to plus as x passes from less than 3 to greater, 

d 2 u . ..... 

while , 2 * s negative on both sides of infinity. Hence we 

infer that z/ is a decreasing function until #=3, and an in- 
creasing one afterwards. Hence a minimum at that point. 
If we substitute for x in equations (1), (2), (3), the num- 
bers 2.3.4. successively we shall have for 

du 2 d 2 u 2 



x- 



x : = : 4 

If we let the ordinates of the curve ABC (Fig. 7) rep 
resent the successive values of u, 
we see that from 2 to 3 the function 
decreases, as is shown by the nega- 
tive value of ^ and at an increas- 
ing rate, as is shown by the negative 

d 2 u 
value of -~ty ; from 3 to 4 the pos- 
itive value of ~ indicates an increasing function, while the 



u—y 9 


dx |/"^ 


dx 2 ty 81 




du 


d 2 u 


u=o 


dx 


dx* - * 




du _ 2 

(IX /\/ 


d 2 u 2 


U ~V 9 


dx 2 " \z~z7 




MAXIMA AND MINIMA. 109 

d 2 u , . 

negative value of , ^ shows that increase to be at a dimin- 
ishing rate. Hence the form of the curve. 

(32) We have seen (Art. 30) that there may be as many 
maxima and minima of a function as there are roots for the 
equation formed by making the value of ^~o. To illustrate 

a case of this kind we take 
Ex. 7. 

U=X^ — 20JT 3 — \^2X 2 — 320^ + 286 (i) 

whence 

du 

(2) 

(3) 

Placing the second member of equation (2) equal to zero, 
we find for x three values as follows : 

X=2 

x=8 

Substituting these values in equation (3) we have for 

d 2 u 



dx 


: 4X S — 


-60X 2 +264^— 


320 


d 2 u 




2 — 120^ + 264 




//-Y?2 


— 123: 





dx % 



--12 



*=5 ^F=-36 

d 2 u 
X=S ~dx^= 12 

from which we infer that for x—2 and x=8 there is a mini- 
mum, and for ^=5 there is a maximum. This will be seen 
by substituting for x in equations (1), (2), (3), successive 
values, as follows 



no 



DIFFERENTIAL CALCULUS. 



x-= 


o 


?l=2&6 


dx 


= - 


-320 


d 2 ii 
dx 2 


-—264. 


X— 


I 


u = 79 


u 


— - 


-112 


a 


= I 5 6 


x= 


2 


»= 3° 


a 


=zr 


O 


a 


= 7 2 


x= 


3 


»= 55 


u 


= 


40 


a 


== 12 


x= 


4 


z/ = 94 


u 


ZZl 


3 2 


a 


= -24 


x= 


5 


z/ = iii 


a 


— 





a 


= -36 


X — 


6 


*== 94 


u 


— - 


- 3 2 


a 


= — 24 


x= 


7, 


»= 55 


a 


= - 


- 40 


a 


— 12 


x= 


8 


»= 3° 


a 


:zz 





a 


= 7 2 


x = 


9 


*= 79 


a 


= 


112 


a 


- 156 


x= 


IO 


^ = 286 


a 


= 


320 


it 


— 264 



If we let the ordinates of the curve ABC (Fig. 8) rep- 



resent the successive values of 
u corresponding to numbers 
substituted for x at the foot of 
each, we see that the function 
decreases at a diminishing rate 
until x=2, when it ceases to 
decrease and begins to increase 
at an increasing rate, as is 



V 



12 3 4-5 e 7 S 9 

Fig. 8. 



du 



shown by the change of sign in -7— and the continued posi- 



dx 



d 2 u 



tive sign of , 2 . But at #=4, although still increasing, as 

du , . .... 

is shown by the positive sign of ~r~ , it is at a diminishing 

d 2 u. . . 

rate, for , 8 is now negative, and thus continues until at 5, 

du 

77— becomes zero, the function ceases to increase and be- 
ax 

gins to diminish at an increasing rate, as is shown by the 



au 



d 2 u 



negative signs of -7— and 7 , 3 at x = 6. But at x=j 



we 



MAXIMA AND MINIMA. Ill 

d 2 u . ... 

have , ., positive, showing that the rate of diminution of 

the function has ceased to increase and begins to diminish, 

until at x=S it has become zero, when the changes at x=2 

are repeated. We notice that between ^=3 and x=4 the 

d 2 u 
sign of , 3 changes from rjlus to minus, showing that be- 



ix 



die 
tween those two points the rate of -7— has changed from an 

increase to a decrease, that is, the function has changed 
from increasing at an increasing rate to increasing at a 
diminishing rate. The exact point where this change takes 

d 2 u ... 

place is where the value of , 2 =0. This will give 

x=$ — V~3 and x=s+Vl> 
which last value corresponds to a point between x = 6 and 
x~J, where the same change is repeated, only in a contrary 
direction. 

From all these cases we deduce the following rule for 
ascertaining the values of the variable that will produce a 
maximum or minimum value for the function, if there be 
any. 

Place the first differential coefficient equal to zero ; and 
substitute each of the roots of this equation for the varia- 
ble in the second differential coefficient. Each one that 
reduces it to a real negative quantity will produce a maxi- 
mum value for the function ; while a similar positive result 
will indicate a minimum. Should any real root thus found 
reduce the second differential coefficient to zero, substitute 
it in the third, fourth, etc., successively, until a real finite 
value is found for some one of the coefficients. If the first 
thus found be of an even order and positive, there will be a 
minimum; if negative there will be a maximum. If the first 
that is reduced to a real finite quantity is of an odd order, 



112 DIFFERENTIAL CALCULUS. 

whether positive or negative, there will be neither maximum 
nor minimum. 

The first differential coefficient may also be placed equal 
to infinity, and if there be any real finite roots, they may be 
treated in the same manner as those obtained by placing it 
equal to zero. In this case, however, a positive sign of the 
second differential coefficient indicates a maximum and a 
negative sign a minimum. 

If a given function contain two variables there must be 
an equation, and one of the variables must be considered as 
dependent on the other. The problem will be to find the 
maximum or minimum value of the dependent variable ; for 
which purpose it must be considered as an implicit function 
of the other, and the differential coefficients will be found 
as in other cases. 

Note. — It may be objected that, herein, the subject of maxima and minima has 
been treated in too prolix a manner, and the reasoning has been unnecessarily repeated. 
I reply, it is of the highest importance that the student should have not only a clear 
and correct, but a familiar, conception of the laws which govern the relations of the 
different orders of rates or differentials, because these are among the fundamental ideas 
of the calculus, and essential to a complete comprehension of the subject. Now unless 
these ideas are presented sufficiently often to render them familiar ; if the student on 
every new occasion is obliged to draw afresh upon his powers of imagination, and go 
through the mental labor of forming his conceptions anew, the study will prove not 
only more difficult, but far less attractive. He will be like a traveler in the dark, who, 
instead carrying a constantly shining lamp to guide his footsteps, must light his candle 
anew for every fresh obstacle. Hence the importance of a full and elaborate explana- 
tion, even at the expense of some, otherwise unnecessary, repetition. 

EXAMPLES. 

Find the value of x for the maximum or minimum value 
of u in the following equations : 

Ex. 8. u=x 3 -\-i8x 2 +105.X. Ans. 

Ex. 9. n~a — bx-j-x 2 . Ans. 

Ex. 10. ^=<3 4 -j-b s x — ex 2 . Ans. 

Ex. 11. u — 3a 2 x 3 — b^x-\-c 2 . Ans. 

Ex. 12. u=a 2 -}-bx 2 — ex 3 . Ans. 



MAXIMA AND MINIMA. 113 

APPLICATION TO PRACTICAL PROBLEMS. 

(33) In order to apply the rules for determining maxima 
and minima of functions to the solution of practical prob- 
lems, it is necessary to obtain an algebraic expression of the 
function, whose maximum or minimum is to be determined, 
in such terms that it shall contain but one variable. No 
specific rules can be given for this purpose, but care must 
be taken to express the function in terms of a variable that 
shall have a range of values beyond that which may be 
required to produce a maximum or minimum, for if it does 
not, although there maybe a kind of maximum or minimum, 
it will not be one in the meaning of the term as used in the 
calculus, as there will be no turning point 'in the value of the 
function, nor any change of sign in the value of the first 
differential coefficient. A few examples will indicate the 
nature of the process more clearly. 

Ex. t. Divide the quantity a into two such parts that 
their product shall be a maximum. 

Let x be one of these parts, then the other will be a— x, 
and the function will be 

x(a— x)= : ax— x 2 
which is to be a maximum. Representing it by u we have 
du 
dx~ =a - 2X W 



(2) 



dx 2 

du 
Placing the value of -j- equal to zero we have 



X =Y 



• • d 2 u 

which the negative sign of , 2 shows to be a maximum. 

Hence when a quantity is divided into two parts their prodicct is 
a maximum when they are equal. 



H4 



DIFFERENTIAL CALCULUS. 



Ex. 2. To find the greatest cylinder that can be inscribed 
in a given right cone. 

Let the height SC (Fig. 9) 
of the cone be represented by 
0, and the radius of the base 
AC by 6, and let x represent the 
distance SD from the vertex of 
the cone to the upper base of 
the cylinder. From the trian- 
gles SAC and SED we have 
SC : AC : : SD : ED or a : b : : x : 

ED, hence 

EBz Jm 

But the area of the upper 
base of the cylinder is Flg - 9 * 




b 2 x 2 



Multiplying this by DC=a—x, the height of the cylinder, 
we have the volume or capacity which we will represent by 
F, and hence 

~b 2 
V= — * x 2 (a — x) 



Now any value of x that will render x 2 (a- 
will render any multiple of it also a maximum, hence - 



-x) a maximum 

a 2 

being a constant factor maybe disregarded in the operation. 
Differentiating twice and representing x 2 (a— x) by u, we 

have 

du 



dx 



-2ax — 2> x 



dx' 
du 



- = 2<z — 6x 



Making the value of ^ equal to zero we have 
x=o and x—-7r 



MAXIMA AXD MINIMA. 115 

The first cannot be a maximum since it reduces the value of 

d~H ... .... 

, 2 to 20, which being positive indicates a minimum. In 

fact, when x=o the cylinder is reduced to the axis of the 
cone, and vanishes with x. The other value x=\a will 

d' 2 u 
solve the problem, since it reduces the value of , 2 to —20, 

a negative quantity which indicates a maximum value for the 
function. Hence 

The maximum cylinder that can be inscribed in a right cone is 
one in which the height of the cylinder is one -third of the height 
of the cone. The radius of the base will also be equal to 
two-thirds that of the base of the cone. The volume of the 
cylinder will be to that of the cone in the ratio of their 
bases, or as 4 is to 9. 

Ex. 3. Required to determine the dimensions of a cylin- 
drical vase, that will contain a given quantity of water with 
the least amount of surface in contact with it. 

Let v represent the given volume of water, x the radius 
of the base of the cylindrical vase, and y its altitude. Then 
we shall have 



from which 



Now the convex surface of the cylinder is equal to 2Txy, 
and substituting the value of y we shall have the convex sur- 
face equal to 

2~ XV 2V 

~x 2 x 

If to this we add the surface of the base =~.# 2 we have the 
whole surface in contact with the water. Calling this surface 
S we have 



V=' 


~x 2 y 




V 


y- 


-x 2. 



n6 



DIFFERENTIAL CALCULUS. 



2V 

S= — + ~x 2 

X 



dx 
d 2 S 



227 



4V 



dx 2 



x° 



+ 2" 



(0 
W 

(3) 



Placing the value of 



whence 






equal to zero we have 



:=\/l 



This value answers to a minimum, since it renders the value 



of 



If we substitute this value of x in the 



d^ p° sltlve - 

expression we found for the value of y, namely 

v 



y— — 



we have 



y 



*</^ 



=<\/I 



hence the minimum surface will be in contact with the water 
when the height of the cylinder is equal to the radius of the base. 

Ex. 4. It is required to inscribe in a sphere a cone which 
shall have the greatest convex surface. 

Suppose the semi-circumference AMB (Fig. 10) to revolve 
about the axis AB, it will describe 
the surface of a sphere, and the 
chord AM will describe the con- 
vex surface of a right cone in- 
scribed in the sphere, and AP will 
be its height, and PM the radius 
of the base. The convex surface 
of the cone, which we will call S, 
will be 




Fig. il,. 



MAXIMA AND MINIMA. 117 

S = 2-PM . J-AM—PM . AM (i) 

We have now to determine PM or AM, either of which 

will determine the other. m 
Let 

AB = 2<2 and AP=^ 



PM =AP . FB=x(2a—x) 



PM=\/ X{2CI—X) 



then 

and 

Again 

AM=\/ 2 ^ 
Substituting these values in equation (i) we have 

S=-PM . AM —T^2ax--X^ . V^^-V 4^2^-2 _ 2A *3 

Differentiating this function we have 

dS_ 4a 2 x—$ax 2 4a 2 — $ax 



dx \/ 4a 2 x2 —iax^ V 4a 2 —lax 

as 

dx ' 



(*) 



Placing this value of -T- equal to zero we have 



In order to determine whether this value of x corresponds 
to a maximum or minimum of the function, it will be neces- 
sary to find the sign of the second differential coefficient. 

Before doing this we will examine a method by which the 
operation may be somewhat abridged. 

We have already seen that when the value of a function 
is reduced to zero by giving a particular value to the varia- 
ble, it does not follow that its differential will also be reduced 
to zero by the same value of the variable (Art. 31), for the 
function in passing through zero may, and probably will, be 
passing from negative to positive, or vice versa, and, there- 
fore, may have a differential or rate of change at that point, 
the same as at any other. Thus the latitude of a vessel on 
the equator is zero, but it may be changing as rapidly there 
as anywhere else. 



Il8 DIFFERENTIAL CALCULUS. 

Now if we wish to obtain the second differential coefficient 
for a particular value of the variable, we may take advantage 
of this circumstance. Suppose we find the first differential 
coefficient to be the product of two or more factors ; either 
of these being reduced to zero will reduce the coefficient to 
zero. In this case we may obtain the value of the second 
differential coefficient for the corresponding value of the 
variable, without differentiating the entire coefficient. For 
suppose we have 

du r jt 

a^yy y 

in which y . y f and y' are functions of x ; this product will 
be reduced to zero by any value of x that will reduce either 
factor to zero. Suppose that for x=a we have y—o; if we 
differentiate the function we have 

<I(jti) d* u _ J{yy' '/) _ // dy y y" d/ yy dy" 

dx dx 2 dx dx dx dx 

and since x~o reduces y to zero, the two last terms of this 

expression become equal to zero, and we have 

d 2 u y y" dy 

ax~ dx 

d 2 u 
hence to obtain the value of ~ry for that value of x that 

reduces one factor of the first differential coefficient to zero, 
we have only to multiply its differential coefficient by those 
factors which do not become zero, and then substitute the 
value of x. If for example we have 

£,^=x(x 2 —a 2 )~x(x+a)(x—a) 
we may reduce it to zero by making 

x=o y x=a or ^ = — a 

d 2 u 

If we wish the value of , for the first value of x. we have 

dx~ ' 

d 2 u 



7 s> — x a -^ a 
ux~ x=o 



MAXIMA AND MINIMA. 



II 9 



If for the second value we have 
d 2 u 

If for the third value we have 



dj*_u 

dx 2 x =- 



-x(x—a) : =2a 2 



Resuming now equation (2) 

dS 4a 2 — $ax __ 



--(40—3*) 

_4a 



dx a^J '4^2 —2ax \/ 40,% — 2ax^ 

which becomes zero by making 3^=4^ or #—-3, and hence 

for that value of x we shall have 

d 2 S a d(Aa—^x) ~3 a 

dx 2 \J <\a 2 — lax ' dx \^4^ 2 — 2ax 

which being negative shows that#=-^ corresponds to a max- 
imum of the function. 

Hence, if a right cone be inscribed in a given sphere it will 
have the greatest possible convex surface when the axis of the 
cone is equal to two-thirds of the diameter of the sphere. 

Ex. 5. A point (Fig. 11) being given within the right 
angle B A C, through which 
a line is to be drawn meeting 
the axes AB and AC, it is 
required to find the distance 
A n such that the length of 
the line between the points 
of its intersection with the 
axes shall be a minimum. 

Let Am=a, Om=b and mn=x 
triangles Omn andpAn give the proportion mn : 0/ 
or 

x: b: : a+x: Ap 
whence 

b(a+x) b 2 (a+x) 2 
Ap= — - — and Ap 2 = ^g 




Then the right angled 
An : Ap 



120 DIFFERENTIAL CALCULUS. 

But 



whence 



whence 



~r~ 2 i ~^-2__ — 2 
Ap -f- An — pn 



__2 b 2 {a+x) 2 , 

pn = \ 2 J +(<r+*) 8 



#+.# 



/#= — r~V^ 2 +* 2 

which is the function of which we are to find the minimum 
value. 

Representing this function by ?/, and considering it as the 
product of two factors, we have 

a-\-x 

du a-\-x x — a 

~dx~ = ~x~~ ' V^l^ +Vd2+X2 ' *> 
Reducing to a common denominator we have 

du (a-{-x)x 2 — a(b 2 -{-x 2 ) x z —ab 2 
dx x 2 ^/ ' b% +x% ~ x 2 A^b* +x2 

Making this equal to zero we have 

To find if this is a minimum we differentiate again, but 
as the numerator of the differential coefficient is equal to 
zero for this value of x (Ex. 4), we multiply its differential 
by 

I 



X 2 \/ b* +x% 

which gives 

d 2 u _ $x 2 3 

dx 2 ~~ x 2 \/W+x~z~ v^ 2 +* 2 
a result that is essentially positive whatever may be the 
value of x. Hence x = ^/ad^ corresponds to a minimum 
length of the line //z. If a and b are equal, we have 



MAXIMA AND MINIMA. 



121 



x=l? or mn=Om 
whence 

Ap=An 

Ex. 6. To find the maximum rectangle that can be in- 
scribed in a given parabola. 

Let ACB(Fig. 12) be 
the parabola of which 
C D is the axis. Let 
D E be the height of the 
inscribed rectangle, and 
let CD=a and CE=^, 
then FE 2 = 2px and 

The area of the rec- ° 

tangle is Fig - I2 * 

FG x ED or 2^/2^(0— x) 
which is to be a maximum. 

Dropping the constant factor 2y/~2p (Ex. 2), representing 
the function by u and differentiating, we have 

die 




w=-ax 



3. 

-x 2 and 



dx 



~-\ax 



.1 ' i 
*-\x* 



which being made equal to zero gives 



a 

x — ■=■ 



hence the altitude of the rectangle is equal to two-thirds that of 
the parabola. 

To show that this is a maximum we differentiate again, 
and- find 



dx* - *** 



-4 



which is negative for every positive value of x, and therefore 



for #=-=- 

o 



Ex. 7. What is the length of the axis of the maximum 
parabola that can be cut from a given right cone ? 



DIFFERENTIAL CALCULUS. 



Let ABC (Fig. 13) be the given cone, 
and FDH the parabola cut from it. 
Let DE be the axis of the parabola, 
and AB the diameter of the base of 
the cone. Represent AB by #, AC by 
/;, and BE by x. Then AE=<2— x, 
FE = \/ax—x-~. Also 

AB : AC : : EB : ED or a : b : : x : ED 
hence 

ED=^ 




But the area of the parabola is equal to 

xb /■ 

2y ax- 



fFHxDE or I a 



which is, therefore, to be a maximum. Dropping the con- 
stant factor ^ (Ex. 2), representing the function by u, and 
differentiating we have 



U — \/ ax3— x± 



du 

dx 



Sax" — 4.x 6 



(1) 

( "* CIX * — AX ^ ) ( 2 ) 

2\ / ax2—x± 2<\/ax3— x ± VJ ' 

Placing the second member of equation (2) equal to zero 
we have x=o and x=-^ ; and differentiating (2) for this last 
value of x we have (Ex. 4) 



d* 



dx 2 



(6ax—i2x 2 ) 



(3) 



2\/ ax%—x± 

Substituting this value of x in the second member of this 
equation it is reduced to 



d 2 u 



dx 2 



-2V3 



-fourths of 



Hence the axis of the maximum parabola is 
slant height of the cone. 

Ex. 8. It is required to determine the proportion of a 
cylinder, that shall have a given capacity, and whose entire 
surface shall be a minimum. 



MAXIMA AND MINIMA. 



I23 



Let a 3 be the capacity of the cylinder, #=|AB (Fig. 14) 

the radius of the base, and y=AC 

be the height ; then the two bases c 

taken together will be equal to 2~x 2 , 

and the convex surface to 2-xxy, so that 

2~(x 2 +xy) is to be a minimum. Now 

a 3 
7:x 2 y=a 3 , whence y = ~z~~E- Substitu- 



ting this value of y, representing the A 

function by u, and differentiating we 

have 

2a 3 
u = 2~x 2 + — 




Fig. 14. 



(1) 







du 2a 3 

dx ^ x 2 




(*) 






d 2 u t 4a 3 x 
dx 2 ~4\ /t_r x ± 




(3) 


Placing 


the value of — equal to zero we 


have 








2 ~x 3 =a 3 =r:x 2 y 






whence 




y=2X 






or the height of the 


cylinder is equal to the diameter of 


the base 


and 




a 

*= 3 , 

V27T 







This value of x corresponds to a minimum value of the 
function, as is shown by substituting it in equation (3), 
which gives 



u 

2 -=I27 



dx 
a positive quantity. 

Ex. 9. To divide a right line into two parts such that 
one part multiplied by the cube of the other shall be a max- 
imum. 

Ans. The part cubed is three-fourths of the given line. 



124 DIFFERENTIAL CALCULUS. 

Ex. 10. To find the greatest right angled triangle that 
can be constructed on a given line as a hypothenuse, 

Ans. The triangle must be isosceles. 

Ex. ii. It is required to circumscribe about a given par- 
abola, a minimum, isosceles triangle. What is the length of 
its axis ? 

Ans. Four-thirds the axis of the parabola. 

Ex. 12. What is the altitude of the maximum cylinder 
that can be inscribed in a paraboloid. 

Ans. Half that of the paraboloid. 

Ex. 13. The whole surface of a cylinder being given, 
how do the base and altitude compare with each other when 
the volume is a maximum ? Ans. 

Ex. 14. Required the minimum triangle formed by the 
axis, the produced ordinate of the extreme point, and the 
tangent to the curve of a parabola. Ans. 



SECTION V. 



APPLICATION OF THE DIFFERENTIAL CALCULUS TO 
THE THEOR Y OF CUR VES. 



SIGNIFICATION OF THE FIRST DIFFERENTIAL COEFFICIENT. 

(34) In order to form such a conception of a line as will 
be adapted to the methods of the differential calculus, we 
must consider it to be the path of a flawing point. 

The law which governs the mcvement of the point deter- 
mines the nature of the line, as to form and position ; and 
this law is expressed in the Cartesian system by the equation 
which shows the relation between the co-ordinates of the 
generating point in every position it may occupy throughout 
its movement. 

The direction in which the point is moving is determined 
by the relative rates at which the coordinates are changing 
their values at the same moment. If the rate of change 
should be constantly the same in each of the coordinates, 
whether negative or positive, the generating point would 
move constantly in the same direction, describing a straight 
line ; and this direction would be determined by the ratio of 
these rates, which in this case would be measured by the 
simultaneous increments or decrements of the coordinates 
themselves. 

I2 5 



126 



DIFFERENTIAL CALCULUS. 




Fig. 15. 



CO 



If, for instance, the coordinates AB and BC (Fig. 15) have 
each a constant rate of increase, 
the ratio of the increments CO and 
OC will be constant, and the gen- 
erating point C will move in a 
straight line, whose direction will 
be determined by the relative rates 
with which those increments are 
produced ; or since the rates, being 
uniform, may be represented by 
the simultaneous increments, the 

direction will be determined by the ratio ^. 

But if, while the rate of change in one of the coordinates 
is constant, that of the other should constantly vary, the 
ratio of their simultaneous increments would be constantly 
changing and the point would describe a curve, whose char- 
acter would be determined by the law which should govern 
these varying rates of change ; and this law would be expressed 
in the equation of the curve. But if the varying rate of change 
in the coordinate should, at any point in the curve, cease to 
vary, and should continue afterwards constantly the same 
as at that point, the generating point would cease to describe 
a curve, and would move in a straight line in the direction 
to which it was then tending; and this direction- would be 
determined by the ratio between the rates of change in the 
coordinates as they existed at the instant they both became 
uniform. 

Now the tangent to any curve is the line which would be des- 
cribed by the generating point if it were to move in the direction 
to which it is tending on its arrival at the point of tangency j 
just as a stone, when it leaves a sling, describes a line tan- 
gent to the curve in which it was moving at the instant. 
Hence the ratio of the rates of change in the coordinates of 



THEORY OF CURVES. 



127 



ne 




any point of a curve determine the direction of the 
tangent to the curve at that point. 

Suppose, for example, that AB (Fig. 16) has a uniform rate 
of increase, while BC has a rate 
of increase that is constantly 
diminishing, the point C would 
describe a curve. 

Now let us suppose that the 
generating point on arriving at the 
point C of the curve should con- 
tinue to move in the direction 
towards which it was then tending, 
and should with a uniform motion, at the same rate as it had 
at C, describe the right line CD, then this line would be tan- 
gent to the curve at the point C. From D draw a line DB r 
parallel to the ordinate, and meeting the axes of abscissas at 
B' ; and from C draw a line parallel to the axis of abscissas 
meeting DB' in O. The triangle CDO will have important 
properties which will require careful investigation. 

From whatever point in the line CD the line DB' is drawn, 
the ratio between the lines CD, CO and DO will be the 
same, and hence as CD is described at a uniform rate, equal 
to that with which the generating point is moving at the 
point C, the lines CO and DO will also be described at a 
uniform rate equal to that with which AB and BC were in- 
creasing at the same instant. Hence the three lines CD, 
CO and OD are the increments that would take place in the 
arc, the abscissa and the ordinate in the same unit of time, at 
their several rates of change existing when the generating 
point of the curve is at C ; and, therefore, CO and OD may 
be taken as symbols representing the rate of increase of the 
abscissa and ordinate of the curve, while CD will represent 
the rate of increase of the curve itself at the point C ; and is at 
the same time tangent to it at that point. 



128 DIFFERENTIAL CALCULUS. 

So that if we designate the length of the curve by s, and 
consider CD as representing ds, we shall have CO=^ and 
OD =dy for the point C of the curve. 

The tangent of the angle which the tangent line CD makes 
with the axis of abscissas is equal to ^-q, and hence, calling 
this angle v, 

2=tang.* (i) 

and since CD =CO w +OD w we have 

ds 2 ~dx 2 -\-dy 2 (2) 

We shall have frequent occasion to use these two equa- 
tions in investigating the properties of curves. 

(35) The usual method of obtaining equation (1) is, to sup- 
pose an actual increment BB' (Fig. 16) given to the abscissa, 
and to find the corresponding increment C'O of the ordinate 
from the equation of the curve. The ratio of these two in- 
crements will not give the tangent of the angle v, which is 
equal to ^7, but will approach it as the increments decrease, 
and when they become infinitesimal or vanish, there is no 
difference between ^q and ^-^. 

This manner of reasoning we have already discussed in 
the introduction to this work, and its defects have been 
shown. We commit an error in giving an actual increment 
to the abscissa and ordinate, for their rates of increase are 
not obtained from any actual increase in value (except where 
the rate is uniform), but from the law of change derived 
from the equation of the curve ; and the suppositive incre- 
ments which we give are not real, but symbolical, repre- 
senting what they would be, by the operation of the law con- 
trolling them at that instant, and are, therefore, a symbolical 
expression of that law. 

The truth is that CO and OD, so far from being infinitely 
small may have any value whatever assigned to them ; so 



THEORY OF CURVES. 



129 



that we may consider them as variables whose simultaneous 
values always correspond to some point in the tangent line. 
In fact, if we differentiate the equation of a curve, and con- 
sider x and y as constants for the point of tangency, dx and 
dy may be considered as the variable coordinates of the tan- 
gent line, with the origin at the point of tangency, and the 
axes parallel to the primitive axes. Under these conditions 
the differential equation of the curve becomes the equation 
of its tangent line. This can easily be shown by a few 
examples. 

Ex. 1. The equation of the circle with the origin at A 
(Fig 17) is 

y 2 =2R^ — x 2 
which being differentiated be- 
comes 

ydy : =('R—x)dx 
If now we consider x and y as 
constants for the point P, and 
dx(=FE) and dy{ = ET) as 
variables, we will replace the 
first by m and /z, and the latter 
by x and y, and we have 

my—(K — n)x 

which is the equation' of the tangent line PT with the origin 
at P, while PE and TE are the abscissa and ordinate of the 
line, and m and n are coordinates of the new origin referred 
to the primitive one, or AB and BP. 

Suppose now we transfer the origin to O, the center of the 
circle. The formulas for transferring to a new origin in a 
system of parallel axes is 

y=b+y f and x=a+x f 
where a and b are the coordinates of the new origin. In 
this case a is equal to BO=AO — AB=R— n, and b is equal 
to PB = — m % and hence by substitution, 




Fig. 17. 



(1) 



13° 



DIFFERENTIAL CALCULUS. 



7/i( — m+/) = (R—n)(R—n+x') (2) 

Calling x" andj/ 7 the coordinates of the point of tangency 
for the new origin, we have 

x"^ — R+7Z or R — ;z = — x" 
also 

y" =BP = ;;/ 

Substituting these values in equation (2) we have 

y (-.y _|_y j _ _ % (_^ ^ % j 

whence 

j/ jy -\-x x — x * -\-y * — R. 4 

or dropping the accents 

yy" -\-xx = R 3 

which is the equation of the tangent to the circle, the origin 
being at the center and x" and y" the coordinates of the 
point of tangency. 

Ex. 2. If we differentiate the equation of the ellipse 
referred to its center and 
axes, we have 

A 2 ydy + B 2 xdx = o 
Making 

y{=BF) and *(=OB) Fig. 18. 

(Fig. 18) constant and dy(=TE) and dx( = PFj) variables, 
and replacing y byy" and x by x\ dx by x and dy by_y, we have 

A 2 y f y +B 3 x x=o 

which is the equation of the tangent line to the ellipse, with 
the origin at P, the point of tangency, and the axes parallel 
to the primitive ones; x" and y" representing the coordinates 
of the new origin referred to the primitive one, and x and y 
the variable coordinates of the tangent line referred to the 
new origin. 

If we transfer the origin back to the primitive one we shall 
have 

x=-a-\-x' and y = -b-\-y t 




THEORY OF CURVES. 131 

where a and b are the coordinates of the new origin — that 
is, the center of the ellipse. But a=—x" and b= — y" (for a 
is essentially positive while x" is essentially negative, and b 
is essentially negative while y" is essentially positive), and 
substituting these values for x and y we have 

Ayy+BW=Ay 2 +B¥ 2 =A 2 B 3 

or dropping the accents 

Ay>+BV.r=A 2 B 3 
Ex. 3. Differentiating the equation of the parabola we 

have 

ydy~pdx 
Representing x, y, dx and dy by x ,y" , x and y respectively, 
we have by substitutior 

y y—px 
for the equation of the tangent line to the parabola, with the 
origin at the point of tangency. If we transfer it to the 
vertex by making y=b+y and x=a+x\ in which b = —y" 
and df= — x\ we have 

-/*+'//*=¥*— jbf. 
in which x" andy" are the coordinates of the point of tan- 
gency for the origin at A. Hence y ,2 =2px ; \ and substitut- 
ing we have 

— 2px" -\-yy' r =-px r — px" 
or, dropping the accents, 

yy —px-\-px —p\x-\-x ) 

which is the equation of the tangent to the parabola at the 
point whose coordinates are x" andj/, the origin being at the 
vertex. 

Ex. 4. Lastly we will take the equation of the Hyper- 
bola referred to its center and asymptotes 

A 2 +B 2 

xy= - 

from which 

xdy-\-ydx=o 



132 



DIFFERENTIAL CALCULUS. 



Replacing y = FB (Fig. 19) by y" , jc=AB by x\ ^ = ET 
by y and ^ = EP by x, we have 

y x-j-x y=o 
in which x" and y" are the coordinates of the new origin 
referred to the primitive one, 
and x and y are the variable 
coordinates of the tangent line 
TP ; the origin being at P and 
the coordinate axes, PM and 
PN, parallel to the assymp- 
totes. 

To transfer the origin to A 
the center of the hyperbola 
make y = b-r-y\ and x= : a+x f J - b being equal to —y" and 
a = — x" ; and substitute these values for x and y. This 




Fig. 19. 



y [x —x ) + x [y —y )— o 

or, dropping the accents 

y" A 2 +B 3 

y—y = — — t\x — x ) oryx -j-xy = 

which is the equation of the tangent line to the hyperbola 
referred to its center and asymptotes. 

Thus it clearly appears that differentials are not infinitely 
small quantities, but are symbols to express the rates or laws 
of variation, which are, in fact, variable functions of the 

GIVEN VARIABLES. 

SIGN OF THE FIRST DIFFERENTIAL COEFFICIENT. 



(36) If x and y represent the coordinates of any curve, 
and while x increases uniformly y should have a positive 
value and, also, increase, its differential will be positive and 
the curve will tend to leave the axes of abscissas in a posi- 
tive direction ; but if y should be decreasing while its value 



THEORY OF CURVES. 



*33 



is positive, its differential will be negative and the curve will 
approach the axis of abscissas on the positive side. 

Again \i y has a negative value and increasing, its differen- 
tial will be negative, and the curve will be receding from the 
axis of abscissas on the negative side ; while if it is decreas- 
ing (being still negative) its differential will be positive, and 
the curve will be approaching the axis of abscissas on the 
negative side (see definition of a differential, Art. 3). 

Hence the following rule : 

When the ordinate and its first differential have the same 
sign the cicj-ve is receding from the axis of abscissas, and when 
they have different signs the curve is approaching that axis. 

Note. — The differential of the independent variable is supposed to be constant and 
positive, and hence the sign of the differential coefficient is the same as that of the 
differential itself of the function. 

This rule may be illustrated by means of the circle (Fig. 
20) whose equation (the origin being at A) is 



from which we obtain 



y6 



a y -. 



zRx — a 



R-x 



~dx 



We see here that from A to C, y and its differential have 
the same sign, and the curve recedes 
from the axis of abscissas. The same 
is true of the curve from A to D. 
But from C to B, and from D to B, 
where x is greater than R, the sign of 
y will be contrary to that of dy, and 
the curve approaches the axis of 
abscissas on both sides. 

We arrive at the same result if we 
consider ^| as representing the tangent of the angle which 
the tangent line makes with the axis of abscissas. From A 
to C and from A to D, where the curve leaves the axis of 




134 DIFFERENTIAL CALCULUS. 

abscissas, the sign of the tangent and of y are alike ; while 
from C to B and from D to B their signs are contrary 

At the point A where y—o> the value of — = ~ x becomes 
infinite, and the curve departs at right angles from the axis 
of abscissas. While at the points C and D the value ^ 

becomes zero, and the curve neither approaches nor recedes 
from the axis of abscissas. And this corresponds with the 
value of the tangent of the angle made by the tangent line 
with the axis of abscissas ; at A and B, —,= oo and the angle 

is a right angle; at C and D, ^=o and the angle is zero; 
the tangent line being parallel to the axis of abscissas. 



SECTION VI. 



DIFFERENTIALS OF TRANSCENDENTAL FUNCTIONS. 

Proposition I. 

(37) To find the differential of a constant quantity raised 
to a power having a variable exponent. 

Let the constant quantity be represented by a and the 
variable exponent by v j then the function will be 



a° 
If we add an increment to v which we will call m we shall 
have 

a*+™=a v a in (i) 

Differentiating this equation we have 

da v+m =a m da v (2) 

and dividing equation (2) by equation^ 1) we have 
da v+m _da v 
a v+m — a v (3) 

This equation being true, irrespective of any particular 
value of m, it will be true for any value we may assign to it ; 
hence the differential of a constant quantity raised to a 
power denoted by a variable exponent, divided by the power 
itself is a constant quantity, or 

da v _ 
a v 
But we have seen (Art. 6) that the differential of the vari- 



136 DIFFERENTIAL CALCULUS. 

able is always a factor in the differential of the function. 

Hence dv will be a factor of C. Calling the other factor 

k Ave have 

C=kdv 

whence 

da* 
-—^—kdv 

or 

da*—a*kdv 

The problem now is to find the value of k. For this pur- 
pose we expand d° by Maclaurin's theorem and have 

a* =A+£v + Cv 2 +Dv 3 +Ev± +etc. 

in which 

dd° d 2 a v d s a v 

A—d°, B—-7- C=~T-; D— -rr-f-etc. 

dv 2dv A 2 . 3 . dv 6 

when v is made equal to zero. * . + 

But we have found 

da* 



hence 



or 



and similarly 



—a*k 

dv 

d[^-\ =da*k=a*k*dv 

d 2 a v 

d*a v _ v . d^a* _ v 4 



from which making v=o we have 

k 2 k* k± 

A = i, B=k, C= — , £>= , £=—-—, etc. 

9 ' 2 ' 2.3' 2.3.4' 

Substituting these values in equation (1) we have 

k 2 v 2 k*v* kW 

a*=i+kv+ + + +etc. 

2 2.3 2.3.4 

and making z'—-r this becomes 



TRANSCENDENTAL FUNCTIONS. 137 

^^^.i-ri+y+a-^+^-^H- etc = 2. 718282 + 

If we represent this number by e we have 
i_ 
a k =-e or a=e k 

If e is made the base of a system of logarithms k would be 
the logarithm of a to that base. 

This was done by Napier, the inventor of logarithms, and 
the system having that base is called the Naperian system. 
We shall indicate the logarithms of that system by the nota- 
tion log., while the logarithms of other systems will be 
noted by Log. We have, therefore, 

da v —a v log. adv 
that is, 

The differential of a quantity raised to a power denoted by a 
variable exponent, is equal to the power multiplied by the Nape- 
rian logarithm of the constant quantity into the differ -ential of 
the exponent. 

Proposition II. 

(38) To find the differential of the logarithm of a varia- 
ble quantity. 

Let the quantity be represented by r and its logarithm by 
v y the base of the system being represented by a. Then we 
have 

r = d° and dr=a v kdv 
whence 

dr 1 dr 

dv= — TTj or d Lo&. r=-7 . — 
a v k d k r 



Representing -^ by M we have 



d Log. r — M ~ 



in which M\s the reciprocal of the Naperian logarithm of 
the base 0, and is called the modulus of the system of loga- 
rithms of which a is the base. Hence 



I38 DIFFERENTIAL CALCULUS. 

The differential of the logarithm of a variable quantity is 
equal to the modulus of the system to which the logarithm be- 
longs, into the differential of the quantity divided by the quantity. 

In the Naperian system the modulus is, of course, one. 
Hence in that system 

d log. r=-p 

from which we learn that in the Naperian system the rate of 
increase of the natural number, whatever may be its value, 
divided by the rate of increase of its logarithm, is always 
equal to the number itself. 

Note. — This principle was used by Napier himself in constructing his table of 
logarithms, and explains his selection of his peculiar base. Hence he is one of the 
first discoverers of t\\o. principle of the differential calculus, although he never applied 
it otherwise than to logarithms. 

(39) If we call e the base of the Naperian system of 
logarithms, a the base of any other system, m the logarithm 
of p to that base, n the Naperian logarithm of p, and s the 
Naperian logarithm of a we shall have 

p—a m p—e 71 a—e 8 

hence 

p— e sm— e ii 

wherefore 

n 1 

n'=-sm or w=— '— — n 

o b 

but — is the modulus of the system, and hence 

The logarithm of a number in any system is equal to the Na- 
perian logarithm of that number multiplied by the modulus of 
the system. 

This property is not peculiar to the Naperian system. 
The logarithm of a number in any system is equal to the 
logarithm of the same number in the common system mul- 
tiplied by the reciprocal of the common logarithm of the 
base of the new system. 

In fact, in any two different systems, the ratio between the 



TRANSCENDENTAL FUNCTIONS. 



139 



logarithms of the same number is constant. Thus let a and 
b be the bases of two systems, m and n two numbers, and x 
and y their logarithms in the first, and u and v their loga- 
rithms in the second system ; then 

?n=-a x n—ay m—b u n—b v 



whence 






a x =#u and a y — b v 


whence 






U V 

a—b x and a— by 



whence 

u v_ 

x ~ y 

or the ratio between the logarithms of the same number in 
two different systems is constant and equal to the ratio be- 
tween the logarithms of any other number taken in the same 
systems. Hence 

log. a : com. Log. a : : log. 10 : com. log. 10 — 1 
whence 



com. Log. a- l ^^==-Mlog. a 



as we have seen. 



Proposition III. 

(40) To find the differential of the sine of an arc. 

Let APD (Fig. 21) be a circle whose center is at O. 
POA be the given angle, then PB 
will be the sine of the arc AP, 
and also an ordinate of the circle 
to the axes OX and OY ; while OB 
will be the cosine of the same 
angle, and also the abscissa of the 
point P of the curve, and AB the 
versed sine of the angle. 

From the equation of the circle 
with the origin at the center we 
obtain 



Let 




Fig. 2 



140 DIFFERENTIAL CALCULUS. 

xdx = — ydy 
dx 2 ~- 



and 

y 2 dy 2 



x* 
If we represent the arc AP by s we have (Art. 34) 

ds 2 =dx 2 +dy 2 

whence 

y 2 dy 2 (x 2 -j-y 2 ) 

ds = ~x~ 2 ~~ +dy2 = x^~ dy 

But 

x 2 +y 2 =R 2 

whence 

R 2 dy 2 
ds 2 =—_ ■£- 



from which we obtain 

~R 



dy=--^rds 



or 

COS. 5" 

^sm.j= — t> — ^ (1) 

that is, 

The differential of the sine of an arc is equal to the cosine of 
the arc into the differential of the arc divided by radius. 

Proposition IV. 

(4 1 ) To find the differential of the cosine of an arc. 

From the equation 

cos. *=VR 2 — sin. 2 s 

we have 

-sin. s.dsm. s 



dcos. s- 



VR 



2 — cm ^ 



sin. s 



Substituting the value of dsin. s (Art. 40) and replacing the 
denominator by cos. s, we have 



TRANSCENDENTAL FUNCTIONS. 141 

— sin. scos.s .ds — sin. s 

dcos.s — ^ = — 5 — ds (2) 

R cos. i- R v ' 

that is, 

The differential of the cosine of an arc is equal to minus the 
sine of the arc into the differential of the arc y divided by radius. 



Proposition V. 

(42) To find the differential of the tangent of an arc. 

From the equation 

R sin. s 

tang. s= 

to cos. s 

we obtain (Art. 14) 

R cos. s . ds'm. s—K sin. s . d cos. s 

dtang. s = 7, * 

G cos/ s 

Substituting for d sin. s and d cos. s their values (Art. 40 

and 41) we have 

R cos. 2 s.ds+K sin. 2 s.ds 

dtang. s— 5 1 — — ■ 

& R cos. 3 s 

or 

(cos. 2 ^ + sin. 2 s)ds R 2 

tftang. s— 2 = T ds (3) 

cos/ s cos. s w/ 

that is 

The differential of the tangent of an arc is equal to the square 
of the radius into the differential of the arc divided by the 
square of the cosine. 

Proposition VI. 

(43) To find the differential of the cotangent of an arc. 
From the equation 

R 2 

cot. j =7 

tang, s 

we obtain 

— RVtang. s 

d cot. s = — 7 o 

tang/ s 



142 DIFFERENTIAL CALCULUS. 

Substituting for dtang. s its value from equation (3) we 

have 

-R 4 ^ -R 2 

dcot.s = g — : —=- — g- ds (4) 

cos/ j - tang." s sin. -5 ^ *v 

that is 
1 7%* differential of the cotangent of an arc is negative, and 
equal to the differential of the arc multiplied by the square of the 
radius, and divided by the square of the sine. 

Proposition VII. 

(44) To find the differential of the secant of an are. 
From the equation 

R 2 

sec. s= 

cos. s 

we have 

— R 2 dcos. s 
d sec. s= i 

cos/ s 

Substituting the value of dcos. s from equation (2) we have 

R sin. s 

dsec.s = 2 — ds (O 

cos/ s VJ/ 

that is 

The differential of the secant of an arc is equal to the differ- 
ential of the arc multiplied by the radius hito the sine, divided by 
the square of the cosine. 

Proposition VIII. 

(45) To find the differential of the cosecant of an arc. 
From the equation 

R 2 

cosec. s=— = 

sin. s 

we obtain 

— R 2 d sin. s 
d cosec. s= = — 5 

sin. *s 



TANGENT AND NORMAL LINES. 151 

and hence 

t 

# y—y —y-(x—x) 

or 

yy +##'=R 2 
becomes the equation of the tangent line to a circle. 

2£r. 2. In the case of the parabola we have 

dy__P 
dx y 
whence 

y-y — y(*-*) 

or 

becomes the equation of the line tangent to a parabola. 

Ex. 3. The equation of the ellipse gives 
dy B 2 x 

dx A 2 y r 

whence 

,_ BV 

or 

A 2 j/+B 2 **=A 2 B 3 

becomes the equation of the line tangent to the ellipse. 

Ex. 4. From the equation of the hyperbola referred to 
its center and asymptotes, we have 

dy___y__ 

dx x 

whence 

r 

t y r t\ 

y-y ——-'{x—x) 

or 

yx +xy — ■ 



iS 2 



DIFFERENTIAL CALCULUS. 



becomes the equation of the line tangent to the hyperbola, 
referred to its center and asymptotes as coordinate axes — 
as in Art. 35. 

Since the normal line is perpendicular to the tangent, if 
a represent the tangent of its angle with the axis of abscis- 
sas, then it will be equal to — — where a represents the tan- 
gent of the angle of inclination of the tangent line. Hence 

, dx 

dy 
and substituting this value in the equation 

y—y — a \x—x ) 
we have 



y—y 



dx 



for the general equation for the normal line, and it may be 
found for any particular curve by obtaining the value of 



dx 



— -r- from the equation of the curve, and making the sub- 
stitution as in the case of a tangent line. 



Proposition I. 



(52) To find the general expression for the length of the 
subtangent to any curve. 

Let AP (Fig. 23) be any curve of which PT is the tangent 
at the point P, TB the subtangent, 
PN the normal, and PB the ordinate ; 
then from the triangle TPB we have 



PB^TB. tang. PTB 



whence 



but 



TB = 



PB 



"tang. PTB 

dy 
tang. PTB=^ and PB=y 




TANGENT AND NORMAL LINES. - 153 

hence 



dx 



TB=j 

that is, 

The subtangent to any curve is equal to the ordinate into the 
differential of the abscissa divided by the differential of the 
ordinate. 

Proposition II. 

(53) To find the general expression for the length of the 
tangent to a curve. 

From the triangle PTB (Fig. 23) we have 

PT= Vpg S +TB 2 

whence 

that is, 

The length of the tangent to any curve is equal to the ordinate 
into the square root of one plus the square of the differential 
coefficient of the abscissa. 

Note. — By the "length of the tangent" is meant that part of the tangent line 
between the point where it intersects the axis of abscissas and the point of tangency on 
the curve. 

Proposition III. 

(54) To find the length of the subnormal to any curve. 
Since the triangle PBN (Fig. 23) is similar to the triangle 

PBT, we have the angle BPN=BTP, and hence 

BN = PB . tang. BPN 
or 

dy 

that is, 



154 DIFFERENTIAL CALCULUS. 

The subnormal is equal to the ordinate into the differential 
coefficient of the ordinate. 

Proposition IV. 

(55) To find the length of the normal to any curve. 
Since p^ 2 (Fig. 23) is equal to p~b 2 +b~n 2 , we have 



PN 



=\Zy*+y^=y\/ 



iy % / dy 2 



dx 2 -sv x ^ dx* 
that is, 

The length of the normal line is equal to the ordinate into the 
square root of one plus the square of the differential coefficient 
of the ordinate. 

Note. — By the "length of the normal" is meant that part cf it which lies be- 
tween the point of its intersection with the axis of abscissas and the point of the curve 
to which it is drawn. 

(56) The following examples will show the application of 
these formulas to particular cases. 

Ex. 1. From the equation of the circle we have 
dx y 

dy x 

hence the subtangent (Fig. 24) is 



-2 



TB=^ 



dx _ y 2 _PB 



dy x BO 

a result that we also obtain 
from geometry. 



Ex. 2. The length of the T 
tangent to the circle is 




TP =V* +-fr=V* +5 =iV x* +y* =5 



We have also by geometry 

TP : PO : : PB : BO 



TANGENT AND NORMAL LINES. 155 

whence 

PO . PB Ry 

TV — =— — 

ir BO x 

Ex. 3. The normal line of the circle is 

Ex. 4, The subnormal of the circle is 

dy x 

Ex. 5. From the equation of the parabola we have 

dx _y 

dy ~~p 

hence the subtangent (Fig. 25) is 

dx y 2 2px 
TB=y- r =^ r =~- = 2X = 2AB 
ay p p 

a result which we have also 
from geometry. 



Ex. 6. The tangent of che 
parabola is 




We have just seen (Ex. 5) that TB = 2AB, hence 

^b 2 = : 4ab' 3 — A x% 
whence 

TP= V y* + 4 x* =Vte 2 + tb g 

as is evident from the figure. 

Ex. 7. The subnormal to the parabola is 

BN=y-f=jA=p 
y dx y 

as we find from geometry. 



i56 



DIFFERENTIAL CALCULUS. 



Ex. 8. The normal to the parabola is 



PN 



-y 



V*-^=yV*+y= v s~+J> 



= Vpb 2 +1 



which is evident from the figure. 



Ex. 9. From the equation of the ellipse we have 
dy__ B 2 x 
dx " A 2 y 
and the subtangent (Fig. 26) is 



dx 



A 2 y 2 



A 2 -: 



BT =^ = - B 2 x 

This value for the subtangent 
does not contain B, and hence 
is the same for all ellipses hav- 
ing the same major axis, the 
abscissa being the same. Hence " Q 

the tangent to the circle at P Fig. 26. 

will intersect the axis of abscissas at T, and 

PB 3 OP 2 -OB S A 2 -x 2 




BT ~OB " 

as we have already found, 



OB 



x 



SECTION VIII. 



DIFFERENTIALS OF CURVES. 

(57) We have seen (Art. 34) that the differential of a 
curve is equal to the square root of the sum of the squares 
of the differentials of the ordinate and abscissa. Hence to 
find the differential of any particular curve, we must find 
from its equation, the differential of one of the coordinates 
in terms of the other. The formula will then give the dif- 
ferential of the curve in terms of a single variable. 

EXAMPLES. 

Ex. 1. From the equation of the circle we have 
xdx xdx 

} '~~~ y ~~Vr 2 -x 2 

hence if we designate the arc by u we have 



<fo=Vdx* +dy' 2 = \/dx 2 + rT 2 ^ 2 



x 2 dx 2 _ R<£* 



which is the differential of the arc of a circle in terms of 
the variable abscissa.. 



Ex. 2. From the equation of the parabola we have 

r 
dx =~:dy 

P 

and calling the length of the arc u we have 

/ / v 2 dv 2 dv / 

du = Vdx 2 +dy 2 = \J dy 2 + Z -TT-=-J-Vp 2 +y 2 

157 



*5* 



DIFFERENTIAL CALCULUS. 



Ex. 3. From the equation of the ellipse we have 
B 8 _ _ B 2 x 



y 2 =-t^(A 2 — jc 2 )and dy = — 7 2 ^ 



hence 



tf> 2 = a ±_.*dx* = 



7 



A^-*;: A ±|!(A 2 -. 2 ) A 8 (A 8 -* 8 ) 



hence 



a 7 u=\/dx 2 +dy % = ttj&c V/ ■ 



A 4 -(A 8 — B 8 )* 8 



DIFFERENTIALS OF PLANE SURFACES. 

(58) Every surface may be considered as generated by 
the flowing of a line. 

If we wish to obtain the rate at which the surface is gen- 
erated we must, if possible, consider every point in the line 
to be moving in a direction perpendicular to the line itself, 
if it is straight, or to its tangent at that point if it is a curve. 
For the only method of estimating the rate at which the 
surface is generated is by means of the length of the gener- 
ating line and the rate with which it moves. Now unless the 
movement is made in a direction perpendicular to the line, 
the rate of its motion will be no criterion of the rate with 
which the surface is generated. Thus the line AB (Fig. 27) 
moving in a direction perpendicular to itself will generate 
the rectangle ABCD, but if it D c 

move at the same rate in any 
other direction as Ad, the surface 
generated in the same time will 
be less until if it should move 
in its own direction it would 
generate no surface whatever, 
the simple movement of the line may be properly 
an element in estimating the rate of generation of the 
surface, it must always be supposed to take place in a 



A B 

Fig. 27. 

Hence in order that 



DIFFERENTIALS OF CURVES. 



159 



direction perpendicular to the line itself at every point. 
Otherwise we must include in our estimate of the rate, the 
sine of the angle made by the line with the direction in 
which it moves, which in most cases would be inconvenient, 
and, in many, impracticable. 

(59) A plane surface may be generated in two ways by a 
straight line — by moving so as to be always parallel to 
itself, or by revolving about a fixed point. If it is supposed 
to be generated by the first method, and the boundary line 
is symmetrical about the axis of abscissas, the ordinate of 
the line is taken as the generatrix, and while it moves par- 
allel to itself one of the extremities is in the line, and the 
other in the axis, and thus half the surface is generated. 

Thus if we consider AB, the diameter of the circle ADB 
(Fig. 28) as the axis of abscissas, we would consider the 
upper half of the circle as generated by 
the ordinate DE moving parallel to it- 
self, one extremity being always in the 
curve and the other in the axis AB. 
And similarly with a surface bounded 
by any other line that is symmetrical 
about the axis of abscissas. 

(60) The differential or rate of increase of any surface 
at the moment the generating line has arrived at any given 
position, such as BC (Fig. 29), will be represented by the 
increment that zuould take place (Art. 2) in 
a unit of time if the surface should in- 
crease uniformly, after the generating line 
should leave the position BC at the same ^ 
rate as that with which it arrived there. 
Now, in order that the increment may be uniform, the gen- 
erating line must maintain the same length and flow at an 
unvarying rate. Thus let AC be the curve and CB the gen- 
erating line of the surface ACB ; and let B<£ represent the 




Fig. 28. 




S b 

Fig. 2 



l6o DIFFERENTIAL CALCULUS. 

uniform increment of AB in a unit of time, at the same rate 
as at B ; then the rectangle CcbT> will represent what would 
be the uniform increment of the surface during the same 
unit of time at the rate at which it was increasing at CB. 
And, hence, if we consider the increment Bb as the sym- 
bol representing the rate of increase of AB, the rectangle 
O^B will be the proper symbol to represent the rate of in- 
crease or differential of the surface ACB. But the rectan- 
gle is equal to BC . Bb j and if we call AB x 9 BC will be y, 
and Bb the differential of x. Hence CcbB, or the differential 
of the surface will be 

ydx 
that is 

The differential of a plane surface bounded by the axis of 
abscissas and a curve, is equal to the ordinate multiplied by the 
differential of the abscissa. 

(61) In order to obtain the differential of any particular 
plane surface we must know the equation of the line that 
bounds it, in order that we may eliminate x or y from the 
formula. We shall then have the differential of the surface 
in terms of a single variable. 

Example i. 

(62) To find the differential of a triangle. 

Let ABC (Fig. 30) be the triangle, referred to A as the 
origin and AB and AD as coordinate axes. C 

The equation of the line AC is 

y=ax 
hence 

ydx—axdx 
which is the differential of the surface of Fig. 30. 

the triangle; a being the tangent of the angle made by. the 
line AC with the axis AB. 




Differentials of curves. 161 

Example 2. 

(63) To find the differential of the surface of a semi- 
circle. 

If we take the equation of the circle with the origin at 
the extremity of the diameter, we have 

whence 

ydx —dx\/ 2 Rx—x 2 
which is the differential of the surface of a semicircle, the 
origin being at the extremity of the diameter. If we take 
the origin at the center we have 

ydx =dx\/ ^2__ x 2 

Example 3. 

(64) To find the differential of the surface of a semi- 
ellipse. 

If the ellipse be referred to its center and axes, we have 
from its equation 



y=-VA 2 — x 2 

hence 

B 



ydx=-rdxV A 2 —x 2 
which is the differential of the surface of the semi-ellipse. 



Example 4. 

(65) To find the differential of the surface of a semi- 
parabola. 

From the equation of the parabola referred to its vertex 
and axis we have 

y— V 2px 



1 62 DIFFERENTIAL CALCULUS. 

hence 



1 



ydx =dx\/ ipx =\/2j>x 2 dx 
which is the differential of the surface of a semi-parabola. 

DIFFERENTIALS OF SURFACES OF REVOLUTION. 

(66) A surface of revolution is one which may be- gener- 
ated by a curve revolving about a line in the same plane. 
Every point in the revolving curve will describe a circle 
whose plane is perpendicular to the axis of revolution and 
whose center is in the axis. Any plane passed through the 
axis will cut from the surface" a curve which is identical 
with the revolving curve. 

Such a surface may also be supposed to be generated by 
the circumference of a circular section, made by a plane 
passed through the surface perpendicular to the axis, mov- 
ing parallel to itself with its center in the axis of revolution 
and its radius varying in such a manner, that its circumfer- 
ence shall always intersect the meridian section or directing 
curve. 

The rate of increase, or differential, will be determined, as 
in other cases, by finding the surface that would be generated 
in a unit of time, if the generating circle were to move dur- 
ing that time, without change of magnitude at a uniform 
rate, equal to that with which it arrived at the point of dif- 
ferentiation. Such a surface would be equal to the circum- 
ference of the generating circle into the line which repre- 
sents its rate of motion. Now the center of the generating 
circle is supposed to move along the axis at a uniform rate, 
hence its circumference will move along the directing curve 
at the same rate as the generating point of the curve ; so 
that the line which represents this rate will be the same as 
the differential of the curve. 

Moreover the suppositive differential surface that we are 



DIFFERENTIALS OF CURVES. 163 

seeking must be generated at a uniform, rate, and hence the 
diameter of the generating circle must not change ; so that 
the surface will be that of a cylinder, whose base is the cir- 
cumference of the generating circle at the point of differen- 
tiation, and its height, the line which represents the differ- 
ential of the directing curve at the same point. 

If now we take the axis of abscissas as the axis of revo- 
lution, the radius of the generating circle will be an ordinate 
of the directing curve and the differential of the curve will 
be Vdx 2 +dy 2 (Art. 34); and hence calling the surface of 
revolution S, we have 

that is 

The differential of a surface of revolution is equal to the cir- 
cumference of the generating circle into the differential of the 
directing curve. 

To apply this formula we obtain from the equation of the 
directing curve, the value of one variable in terms of the 
other, and by substitution obtain the differential in terms of 
a single independent variable. 

Example i. 

(67) To find the differential of the surface of a cone. 

In this case the revolving line is straight, and not a curve, 
but the principles of the rule apply 
equally well. 

Let AC (Fig. 31) be the revolving ele- 
ment of the cone, and AB the axis of * 
revolution and of abscissas, the origin 
being at A. Then we have for the equa- Fig. 31 . 

tion of the line AC,y=ax and dy—adx, a being the tangent of 
the angle BAC. 

Substituting these values in the formula we have 
dS = 27zax\ / a 2 dx 2 -j- dx z =27zaxdx\/ a 2 + l 




164 differential calculus. 

Example 2. 

(68) To find the differential of the surface of a sphere. 

From the equation of the circle we have 

xdx x 2 dx 2 

dy= — and dy 2 = — 5 — 

hence 



2 ~yVdx 2 +dy 2 = 2 -y\J X / dx * 



x 2 +j 
*"j'v ax" -\-ay =2~y/y 

whence 

dS = 2-Rdx 
for the differential of the surface of a sphere. 

As the entire expression besides dx is composed of con- 
stants, we infer that the surface of a sphere increases at the 
same rate as the axis. 

Example 3. 

(69) To find the differential of the surface of a parabo- 
loid of revolution. ' 

From the equation of the parabola we have 

ydy y 2 dy 2 

dx= — : — and dx 2 = — 7^ — 
/ p* 

hence 

d§= z 2~yVdx 2 +dy 2 = 2 -y\/ y / Jy 2 

Example 4. 

(70) To find the differential of the surface of an ellipsoid 
of revolution. 

We found (Art. 57) that the differential of the elliptic 
curve is 



1 7 4 /A^-(A 2 -B 2 )s 



A 2 -x 2 
hence if we substitute this expression in place of V dx 2 +dy 2 



DIFFERENTIALS OF CURVES. 165 

in the formula, and for y its value derived from the equation 
of the ellipse, we have 

/B 2 , , „ Z 1 , /A 4 -(A 2 -B 2 W 2 

which becomes by reduction 

2-B^/x / - 

dS=—^ r VA±-(A 2 -B 2 )x 2 



DIFFERENTIALS OF SOLIDS OF REVOLUTION. 

(71) A solid of revolution is one which is described or 
generated by a surface, bounded by a line and the axis 
about which it revolves. If this axis be that of abscissas, 
then the ordinates of the bounding line will describe circles, 
of which they will be the radii and the centers will be in 
the axis. Any one of these circles may be considered as 
the generatrix, which describes the solid by moving parallel 
to itself, as in the last case. But it is now the surface of the 
circle and not merely its circumference that generates ; and 
its movement is measured along the axis, the rate being the 
same as that by which the abscissa of the directing curve is 
increasing. 

Now the rate of increase of a solid of revolution is meas- 
ured by a suppositive increment that woicld be described in a 
unit of time, by the generating circle moving uniformly 
along the axis, with its diameter unchanged at the same 
rate as that with which the abscissa is generated. Hence 
such a solid would be a cylinder whose base is the genera- 
ting circle, and whose altitude is the line representing the 
differential of the abscissa. But the area of the generating 
circle is 7ty 2 y and the altitude of the cylinder is dx ; hence 
the cylinder representing the differential of a solid of revo- 
lution, would be expressed by the function, 

~y 2 dx 



l66 DIFFERENTIAL CALCULUS. 

hence, 

The differential of a solid of revolution is equal to the gener- 
ating circle multiplied by the differential of the abscissa of the 
bounding line. 

Example i. 

(72) To find the differential of the volume of a cone. 

If we take the vertex of the cone for the origin, and the 
axis of abscissas for its axis, the equation of the revolving 
line will be 

y—ax 

and hence calling v the volume of the cone we have 

dv — ~y 2 dx =~a 2 x 2 dx 
in which a is the tangent of the angle made by the revolv- 
ing line with the axis, and x the distance from the vertex to 
the base of the cone, 

Example 2. 

(73) To find the differential of the volume of a sphere. 
If we take the origin at the extremity of the diameter, the 

equation of the revolving semi-circle will be 

y 2 =2~R.x—x 2 
in which R is the radius of the sphere, and x any portion of 
the axis of revolution measured from its extremity at the 
origin until it equals 2R ; hence the formula for the differ- 
ential becomes 

dv= : xy 2 dx=-(2'Rx— x 2 )dx 

Example 3. 

(74) To find the differential of the volume of an ellipsoid 
of revolution. 

If we suppose the semi-ellipse to revolve about its major 



DIFFERENTIALS OF CURVES I 67 

axis, it will generate an oblong ellipsoid of revolution, oth- 
erwise called a prolate spheroid. If we take the origin at 
the extremity of the transverse axis, the equation of the 

ellipse is 

B 2 

y 2 — ~\2{ 2 -^ x ~~ x2 ) 

and hence the formula for the differential of the volume 

becomes 

B 2 
dv=ny 2 dx= = 'n-r%(2Ax — x 2 )dx 

in which A is the semi-transverse and B the semi-conjugate 
axis of the ellipse which generates the ellipsoid of revolu- 
tion. 

If we take the conjugate axis of the ellipse for the axis 
of revolution and its extremity for the origin, we have 

A 2 

y 2 =-^{2Bx-x 2 ) 

and 

A 2 

dv=--=r^(2~Bx— x 2 )dx 

In this case the volume is an oblate ellipsoid, or other- 
wise, an oblate spheroid. 

Example 5. 

(75) To find the differential of the volume of a parabo- 
loid of revolution. 

The axis of the parabola being the axis of revolution, and 
the origin at the vertex, we have 

dv = ~y 2 dx = 2 -pxdx 
in which / is the parameter of the revolving parabola that 
generates the volume. 



SECTION IX. 



POLAR CURVES. 



Proposition I. 

(76) To find the tangent of the angle which the tangent 
line makes with the radius vector. 

Let CC (Fig. 32) be any curve of which we have the 
polar equation. Let P be the pole ; PM=r the radius vec- 
tor; P£ = R, the radius of the 
measuring arc bd ; b~Pd, the vari- 
able angle =vj and OT 3 the tan- 
gent to the curve at the point 
M. Produce the radius vector, 
PM to R, draw RO perpendic- 
ular to PR, meeting the tangent 
in O ; draw ON parallel to RM, 
and MN parallel to RO, meet- 
ing each other in N. Join PN, 
and draw ab parallel to MN meeting PN in a. Suppose the 
radius vector to revolve around the point P in the direction 
from d towards b. 

The generating point, being supposed to have arrived at 
the point M of the curve will be subject to two distinct, 
although mutually dependent, laws or tendencies. One of 
these tendencies arises from the law of change in the length 




Fig. 32. 



POLAR CURVES. 169 

of the radius vector, which causes the generating point to 
move outward in the direction of its length. The other ten- 
dency arises from the revolving motion of the radius vector 
which causes every point in it (including, of course, the 
generating point) to move in a direction perpendicular to 
itself. Hence in this case, the law would incline the gen- 
erating point to move in the direction MN. If then we 
take the distance MR to represent the uniform outward 
movement that would take place, under the influence of the 
first law in a unit of time, it will represent the rate of 
change in the radius vector arising from that law, and is, 
therefore, the symbol of that rate ; that is 

If we take MN to represent what would be the uniform 
movement under the second law in the same length of time, 
it will represent the rate with which it tends to move in the 
direction MN arising from that law. Now as both these 
laws act together without disturbing each other, the gener- 
ating point, if left to its tendency at the point M would move 
in such a direction as to obey both laws or influences at the 
same time ; and hence at the end of the same unit of time 
would be found at O, having described the line MO ; the 
departure from the line MN being ON=MR, and the depar- 
ture from the line MR being RO^MN. But the generating 
point of a curve, if left to its tendency at any time would 
move in a line tangent to the curve, and since the line MO 
would be uniformly described in a unit of time, it represents 
the rate of increase of the curve, and is also tangent to it. 
Hence if we call the length of the curve u we have 

yiO—du 

The point b at the intersection of the radius vector with 

the arc of the measuring circle, tends to move in the direction 

ba, and if left to that tendency would describe that line in 

the same time that the generating point would describe the 



170 DIFFERENTIAL CALCULUS. 

line MN; for the rate of movement of b is to that of M as 
Tb is to PM, or as ba is to MN. If then we consider b as a 
point in the arc of the measuring circle, we may consider 
betas representing its rate of increase, that is the rate of in- 
crease of the angle b~Pd, and hence 

ab=dv 
But from the triangles PMN and Tba we have 

Vb : PM : : ab : MN . 
hence 

T PM . ab rdv 

Now the tangent of the angle PMT^MON is equal to 

MN_ MN 
R NO ~ R MR 
and substituting the value of MN just found and of MR, 

we have 

rdv 
Tang. PMT=-t- 
to dr 

that is 

The tangent of the angle which the line tangent to a polar 
curve makes with the radius vector is equal to the radius vector 
into the differential of the measuring angle divided by that of the 
radius vector. 

(77) Since MNO (Fig. 32) is a right angled triangle, we 
have 

MO 2 =MN 2 +N0 2 =MN* +MR 3 
hence by substitution 

r 2 dv % 



whence 



or making R = i 



dti 2 — vz +dr 2 



du=-^Vr 2 dv 2 + R 2 dr 2 



du = x/ r 2j v 2 + d r 2 



POLAR CURVES. 



171 



that is 

The differential of the arc of a polar curve is equal to the 
square root of the sum of the squares of the radius vector into 
the differential of the measuring angle, and of the differential 
of the radius vector. 

Proposition II. 

(78) To find the subtangent of a polar curve. 

The subtangent of a polar curve is the projection of the 
tangent on a line drawn through the 
pole perpendicular to the radius vector 
of the point of tangency. 

Hence if PT (Fig. 33) be drawn per- 
pendicular to PM, meeting in T the 
tangent to the curve at the point M, 
then PT will be the subtangent. Since 

MP . tang. PMT 

PT = 

R 

we have by substitution 




Fig. 33. 



r 
PT =R 



rdv r 2 dv 



dr Kdr 

that is 

The subtangent of a polar curve is equal to the sqitare of the 

radius vector into the differential of the measuring arc divided by 

R into the differential of the radius vector. If we make R=r 

rdv 
-—7- =tangent of PMT. 



we have PT : 



Proposition III. 



(79) To find the value of the tangent to a polar curve 
The tangent to a polar curve is that part of the tangent 

line which lies between its intersection with the subtangent 

and the point of tangency. 

Hence MT (Fig. 33) will represent the tangent, and since 



172 DIFFERENTIAL CALCULUS. 

MT 2 =PM 2 +~PT 2 
we have 

—-2 o r^dv 2 

MT ='»+Ri*i 

or 



,^ m . / r 2 dv 2 r , — 

MT =r\f 1 + ^-r- 3 = j^V R »<*■» +r W 



or making R = i 



MT=r\/i 



r 2 dv 2 
dr 2 



Proposition IV. 

(80) To find the subnormal to a polar curve. 

The subnormal of a polar curve is the projection of the 
normal line on a line drawn through the pole perpendicular 
to the radius vector for that point of the curve to which the 
normal is drawn. 

Hence if MB (Fig. 33) be a normal at the point M, BP 
will be the subnormal. 

The triangles MBP and MTP being similar, the angles 
MBP and PMT are equal, and since 

tang. MBP 



we have 



or 



PM =BP- 



rdv 
r=BP^ 



Rdr 
Rdr 



dv 
that is 

The subnormal of a polar curve is eqital to radius into the dif- 
ferential of the radius vector, divided by the differential of the 
measuring arc. 

(81) The normal line MB (Fig. 33) is equal to 

VmfHpb 2 



POLAR CURVES. 173 

or 



MB=\/r 



K 2 dr 2 



dv 2 

(82) While the point at the extremity of the radius vec- 
tor describes the line of a polar curve, the radius vector 
itself generates the surface bounded by the curve. 

Now the point M of the line PM (Fig. 32) tends to move in 
the direction MO, and every other point in the line PM will 
tend to move in a direction parallel to MO, and at a rate 
proportional to its distance from the fixed point P. Hence 
if the point M were to be found at O, the line PM would 
assume the position P.O, and the triangle PMO would be 
that which would be generated at a uniform rate by the 
radius vector PM if left to its tendency when in that posi- 
tion, so that the triangle PMO is the true symbol to repre- 
sent the rate at which the surface bounded by the polar 
curve is generated, or, designating the surface by O, we have 
triangle PMO=</0 

But since ON is parallel to MP, we have 

triangle PMO=triangle PMN 

and 

PM x MN 
PMN= . 

2 

and substituting here the values already found for these 
terms, we have 

' r 2 dv 

dO=—^- 

2R 

hence 

The differential of a surface bounded by a polar curve is 
equal to the square of the radius vector into the differential of 
the measuring arc divided by twice its radius. 

SPIRALS. 

(83) If a right line revolve uniformly in the same plane 
about one of its points, and a second point should at the 



174 



DIFFERENTIAL CALCULUS. 



same time approach to, or receded from the fixed point, 
according to some prescribed law, it would generate a curve 
called a spiral. 

The fixed point is called the pole, and the curve generated 
during one revolution of the line is called a spire. There 
being no limit to the number of revolutions of the line, the 
number of spires is infinite, and a line, drawn through the 
pole will intersect the curve in an infinite number of points. 

Hence there can be no algebraic relation between the 
ordinates and abscissas of the curve, and its conditions 
must be expressed by a polar equation which will be in the 
form 

in which r is the radius vector and v the measuring arc of 
the variable ande. 



SPIRAL OF ARCHIMEDES 

(84) This spiral is one in which the radius vector is con- 
stantly proportional to the corresponding arc which measures 
its angular movement. Hence its equation will be 

r=av (i) 

The curve may be constructed in the following manner. 
Divide the circumference of 
the measuring circle into eight 
equal parts by the radii AB, 
AC, AD, AE, etc, (Fig. 34) ; 
also the radius AB into the 
same number of parts. Then 
lay off from the center one of 
these parts on AC, two on AD, 
three on AE, and so on, there 
being eight on AB, nine on 
AC, ten on AD, and so on. 
Through the points thus found draw the curve commencing 




POLAR CURVES. 175 

at the pole. The radius vector will be to the corresponding 
measuring arc as the radius of the measuring circle is to the 

circumference ; or a will be equal to — -. 

Note. — In this construction we have supposed the radius of the measuring circle to 
be equal to the radius vector after one revolution. Of course any other proportion 
might be taken, but as the magnitude of the spiral does not depend on - that of the 
measuring circle, the radius of the latter may always be taken equal to the radius vec- 
tor after one revolution. 

If we differentiate equation (1) we have 

dv=-adv (2) 

In a polar curve (Art. 76) the tangent of the angle which 
the tangent line makes with the radius vector is equal to 

dv 
dr 
and from equation (2) we have 

dv _ 1 
dr a 

hence the tangent of the angle APT is equal to ■£, or in this 

case, to 

2~r 

This tangent will after one revolution be equal to 

2-R 
(85) The subtangent of a polar curve (Art. 78) is 

r*dv 

Rdr 
which becomes for this curve 



Ra R 

or, making r=R, we have 

subtangent = 2~R 
equal to the tangent of the angle made by the tangent line 
with the radius vector ; and also equal to the circumference 
of the circle described by the radius vector as a radius, 



176 DIFFERENTIAL CALCULUS. 

when the point of tangency is at the circumference of the 
measuring circle. 

If we make ^=/Z2-R, that is, if the tangent be drawn to 
the curve after n revolutions of the radius vector, then 



whence 



v n2~R 



dv 1 ?Z2~R r' z dv 

and -o , =/Z27ir 



dr a r Rdr 

that is 

After n revolutions of the radius vector, the subtangent 
is equal to n times the circumference of a circle described 
by the radius vector as a radius. 

For the subnormal whose value is (Art. 80) 

Rdr 
dv 
we have 

dr r 

dv v 

hence 

Rr 

subnormal = — ■ 

v 

If the normal is drawn at the point B then 

and we have 

r 

subnormals — ' 

27Z 

that is 

The subnormal is equal to the radius of a circle of which 
r=R is the circumference. 

THE HYPERBOLIC SPIRAL. 

(86) The equation of the Hyperbolic Spiral is 

rv=ab 



POLAR CURVES. 



177 




Fig. 35. 



in which r is the radius vector, v the measuring arc, a the 
radius of the measuring circle and b the unit of the meas- 
uring arc — ab being, of course, a constant quantity. It is 
called a Hyperbolic spiral because its equation resembles that 
of a hyperbola referred to its center and asymptotes. 

To construct this curve describe a circle with a radius 
PA (Fig. 35) 
equal to a. 
Lay off from 
A an arc 
AB=£as the E 
unit of the 
measurin g 
arc z>, and 
continue this 

. division around the circumference of the measuring circle. 
Also lay off A^^-g-^, Ar=^b, and so on. 

Through these points of division in the circumference 
draw the radii PB, PC, PD, PE, and so on, and produce the 
radii Fs and Pr. On these radii lay off Yc=-\a, Yd=\a, 
Ve=^a, and soon ; also TO — 2a, PQ=4<z. Draw the curve 
through the points thus found. 

The radius vector multiplied by the measuring arc, count- 
ing from A, will always be equal to the radius of the meas- 
uring circle into the unit of the arc, that is 

rv=ab 

We see from the equation that r increases as v diminishes, 
and vice versa. If v—o r becomes infinite, and hence the 
radius vector, through A, will never reach the beginning of 
the curve. If r=o then v will be infinite, hence the curve 
will never reach the pole. 

If we take any point O in the spiral and join OP, then 
OP will be equal to r, and the arc As=v. Draw OR per- 
pendicular to PA and we have 



T78 DIFFERENTIAL CALCULUS. 





OR = 


OP . sin. As r sin. v 




PA ~~ a 


hence 




OR. a 






r — . 

Sill. V 


and since 




rv — ab 


we have 




OR . av 

: = z ab 

sm. v - 


whence 




sin.z> 
OR = b 



V 

As sin. v is always less than v, the line OR will always be 
less than b, but may be made to approach that value as near 
as we please. Hence if we draw a line MN parallel to PA, 
at a distance from it equal to ^=arc AB, it will be an asymp- 
tote of the curve. 
Since 

dv __ v 
dr r 

the subtangent (Art. 78) 

r 2 dv rv 

R^/r a 

and since 

rv=ab 
we have 

subtangent =— b=- — arc AB 
a constant quantity. Thus Vm or P/z^AB or PM 

Also the tangent of trie angle made by the tangent line 
with the radius vector (Art. 76). 

rdv 

that is, ar 

The tangent of the angle which the tangent tine makes with the 
radius vector is negative and equal to the arc which measures the 
angle made by the radius vector with the fixed line PA. Hence 



POLAR CURVES. 



179 



this angle is obtuse on the side of the radius vector toward 
the origin, while the subtangent, being also negative, lies on 
the side opposite to the origin. 



THE LOGARITHMIC SPIRAL. 

(89) The equation of the logarithmic spiral is 
z^ = Log. r 
in which v represents the measuring arc and r the radius vector. 

The equation 
may also be put 
into the form F 

a v = : r 
the relation be- 
tween v and r be- 
ing such that 
while v increases 
in arithmetical 
progression rwill 
increase in geo- 
metrical progres- 
sion. Hence the 
curve may be con- 
structed . by lay- 
ing off on the Fig. 36. 
measuring arc the equal distances At?, be, cd, de, and so on 
(Fig. 36), and through the points of division drawing the 
radii Yb, Vc, Yd, Ye, and so on, producing them if necessary. 
On these radii lay off the distances PB, PC, PD, PE, and so 
on in geometrical progression, so that 

PB_PC_PD_PE 

PA~PB~PC~PD 
and through the points thus found draw the curve. That 
part of the curve within the circle will be found by laying 
off on the radii Yq, Yp, To, and so on, distances from P by 
the same rule, and thus points of the curve may be found. 




- — : t=7 : f;= I: ft7^— ttr and so on 



ISO DIFFERENTIAL CALCULUS. 

If we make the radius of the measuring circle equal to i, 
and reckon the arc v from the line PA, then the curve will 
pass through the point A, for when v=o we have 

fog. r=0=log. i 
and if we call a the ratio between PA and PB, we shall 
have 

PB = a, PC=^ 3 , PD=# 3 , PE=«±, etc. 
when the exponent is always equal to the number of divis- 
ions of the measuring arc, and is therefore represented by 
the arc itself corresponding to the radius vector, whence 
a v ~r or z^=Log. r to the base a. 
If we differentiate the equation of this curve we- have 

dr 
dv=U— ( 2 ) 

whence (Art. 76) 

rdv rMdr 

tang. PFT=— 7-=— r- = M 

° dr rdr 

that is 

The tangent of the angle made by the tangent line with the 
radius vector is constant and equal to the modulus of the system 
of logarithms to which ilte system belongs. If the system is 
the Naperian, M = i and the angle PDT is equal to 45 °. 

The formula for the subtangent of a polar curve (Art. 78) 

is 

r 2 dv 



Rdr 

dr 



and substituting in this the value of £, from equation (2) we 



have (R being 1) 

sub tan. = — rM 

r 

If M=i then subtang. =r. 

For the value of the tangent we have (Art. 79) 



tang. = Vr 3 +r 2 M 2 = r V 1 +M 3 

If M = i then 

tang=rV 2 



POLAR CURVES. l8l 



For the subnormal we have (Art. 80) 

dr r 

subnormal =-7- =t> 

av M 

If M = i then 

subnormal =r 

For the value of the normal (Art. 81) we have 



normally ^ 2 +-^=ry 1 +^- 3 

If M = i then 

normal — r\/ 2 

These values show that these lines are all in direct pro- 
portion to the radius vector. The same result flows from 
the constancy of the angle made by the radius vector with 
the tangent line. For all the triangles formed by the radius 
vector, the tangent, and the subtangent will be similar to 
each other, at whatever point of the curve the tangent may 
be drawn. The same may be said of the triangles found by 
radius vector, normal and subnormal. Hence these lines 
will always be in proportion to the radius vector. 

To construct a logarithmic spiral for a given base, des- 
cribe a circle with a radius equal to a unit of the radius 
vector, PA, and lay off the arc Kb equal to a unit of the 
measuring arc. Draw the radius vector PB equal to the 
given base; A and B will be points of the curve. Other 
points may be found as already described. 

That part of the curve below the line PA corresponds to 
the negative value of z>, and for that we have 

1 
~~ d° 

in which when r=£>, v will be infinite. Hence the curve is 
unlimited in both directions, 



SECTION X. 



. ASYMPTOTES. 

(88) An asymptote to a curve is a line, which the curve 
continually approaches, but never meets. Such a line is 
said to be tangent to the curve at an infinite distance, by 
which we are to understand that the point of contact to 
which the lines approach is beyond any finite limit. 

That this may be the case it is necessary that, at least, 
one of the coordinates of the curve may have an unlimited 
value. Hence when we are seeking an asymptote to a 
curve, our first inquiry must be, whether the equation of the 
curve will admit of such values for the coordinates or either 
of them. If not, there can be no asymptotes. If it will do 
so for either coordinate, we must substitute that value in the 
equation and ascertain the resulting value for the other 
coordinate. If this resulting value is finite, there is an 
asymptote parallel to the axis of the infinite coordinate ; if 
zero then the axis of the infinite coordinate is itself the 
asymptote. But if it should be infinite, then we must resort 
to the following method. 

Find from the equation the values of the coordinates at 
the points where the tangent line intersects the axis, that is, 

182 



ASYMPTOTES. 



183 



the distances from the origin. 
These points may be found as 
follows : 

Let A (Fig. 37) be the origin of 
coordinates for the curve SO, and 
let PB be tangent to the curve at 
the point P, of which the coordi- 
nates are x r and y . The equa- 
tion of this tangent line is 

y-y = ^ x - x ) 




If we make y =0 we have 



x=x 



dx 



If we make x=o we have 



y—y —x 



ix 



=AD 



If EC be an asymptote, and the values of x and y are 
made such as to remove the point of tangency to an infinite 
distance, then AB and AD will become AC and AE. 

If in such case we have finite values for these distances, 
then there will be one or more asymptotes ; if there is but 
one finite value, there will be one asymptote parallel to the 
axis of the infinite coordinate. If one be zero then the axis 
of the infinite coordinate is it-self the asymptote. If both 
be zero then the asymptote passes through the -origin; but 
if both be infinite there is no asymptote. 

EXAMPLES. 



Ex. 1. The equation of the hyperbola referred to its cen- 
ter and asymptotes is 

xy=M 

in which if x is made infinite y becomes zero ; and if y -is 



184 DIFFERENTIAL CALCULUS. 

made infinite x becomes zero ; hence both axes are asymp- 
totes 

Ex 2. If we consider the hyperbola as referred to its 
center and axes, its equation is 

A 2 j/ 2 =B 2 * 2 -A 2 B 2 
where either x or y may be made infinite, and such value 
makes the other infinite also. Hence we take the formulas 
for the points of intersection of the tangent with the axes, 
which give 

, ,AV_ AV 8 -BV 2 A 2 

*-* -y B3 ^-- B 2^ ~ x > 

and 

. ,B 8 *' AV 2 -BV 2 B 2 
^ ^ A*y A*y y 

both of which values becomes zero, when .a/ andy are made 
infinite. Hence the asymptotes pass through the origin. 

Ex. 3. The equation of the parabola 
y 2 : =2px 
shows that x and y both become infinite together, and hence 
we take 

_ r fit r y 2 

x—x —y~Ti =z x — — — = — x 
J dy p 

and 

_ r r4/ __ f P f _y 

y—y —x -j-, — y —~x — — ■ 
y ' ax J y 2 

both of which values become infinite when x r and y are 

infinite, and hence there is no asymptote to the parabola. 

Ex. 4. If we take the ellipse whose equation is 
A 2 y 2 +B 2 x 2 =A 2 B 2 

we see that neither x nor y can ever be infinite ; in fact y 
can never exceed B nor x exceed A ; hence there is no 
asymptote to the ellipse. 

Ex. 5. The equation of the logarithmic curve is 
x—\og.y 



ASVMPTOTES. 



'% 



It may be constructed by laying off on the axis of abscis- 
sas (Fig. 3S) the distances AB, AC, AD, etc., in arithmeti- 
cal progression, and, on the 
corresponding ordinates, the 
distances Aa, B£, Cc, Dd etc., 
in geometrical progression, 
and drawing a curve through 
the points thus found. We 
see from the equation that if 
either x or y is infinite on the 
positive side, the other will be 
infinite also. 




A B C D E F 

Fig. 38- 

If we apply the formula for the intersection of 
the tangent line with the axis we have (Art. 38) 



y=y — # 



,dy' 



dx 



• , ,y _ // x \ 



(1) 



and 



M 






-x —y ■ 



-—x— M 



(2) 



r dx_ 
~ y dy' ' y 

We see from these values, that when x is infinite x will 
be infinite positively, and y negatively. Hence there is no 
asymptote on the positive side of x. But if x be made 
infinite negatively, y will become zero ; for the logarithm of 
is negative infinity, which shows that the axis of abscissas 
is an asymptote on the negative side. The value of y how- 
ever in equation (1) becomes -ooc ? which is indefinite. 

We learn from equation (2) that the tangent always inter- 
sects the axis of abscissas at a distance equal to M on the 
negative side of the ordinate of the point of tangency. 
Hence the subtangent is constant and equal to the modulus 
of the system to which the curve belongs. If x ~— M, then 
x and y both become zero, and the tangent passes through 
the origin. 

If we put the equation into the form 

y=a x 



l86 DIFFERENTIAL CALCULUS. 

and make x negative it becomes 

i 

which makes y=o when x= oo ; whence we infer that the 
axis of abscissas is an asymptote to the curve on the nega- 
tive side, as already shown. 



SECTION XI. 



SIGNIFICATION OF THE SECOND DIFFERENTIAL 
COEFFICIENT. 



SIGN OF THE SECOND DIFFERENTIAL COEFFICIENT. 

(89) We have seen (Art. 36) that the first differential of 
the ordinate indicates by its sign whether the curve is leav- 
ing or approaching the axis of abscissas ; and by its value it 
determines the rate of such approach or departure ; that is, 
the tangent of the angle made by the tangent line with the 
axis of abscissas. 

As the point of tangency moves along the curve, the rate 
of its approach to, or departure from, the axis of abscissas 
is constantly changing, and upon the rate of this change will 
depend the direction and amount of curvature of the curve. 

Wherever the curve is situated with reference to the axis 
of abscissas, if its rate of departure is an increasing rate, or 
• its rate of approach is a decreasing rate, then the curve is 
convex toward the axis of abscissas ; while if its rate of 
departure is decreasing, or its rate of approach is increasing, 
it will be concave toward that axis. 

(90) Now the second differential of the ordinate will 
determine by its sign whether the first is an increasing or 
decreasing function. If the latter is positive and increas- 
ing, or negative and decreasing, its rate of change (that is 

187 



iSS DIFFERENTIAL CALCULUS. 

the second differential of the ordinate) will be positive (Art. 
3) ; but if it is positive and decreasing, or negative and 
increasing, its rate of change is negative. 

Note. — It will be remembered that the sign of the differential and that of its coeffi- 
cient are always the same, since the differential of the independent variable is always 
uniform and positive. 

(91) If, therefore, the second differential coefficient should 
be positive, the first must be either an increasing positive or 
a decreasing negative function (Art. 3). If the curve is on 
the positive side of the axis of abscissas, it is convex to that 
axis ; if on the negative side it is concave. 

(92) If the second differential coefficient is negative, the 
first must be either an increasing negative function, or a de- 
creasing positive one. Hence the curve, if on the positive 
side of the axis of abscissas will be concave, and on the 
negative side convex to that axis. 

(93) To illustrate these principles let us suppose the 
second differential coefficient to be positive, then the first 
must be a positive increasing 
or a negative decreasing func- 
tion. The curves in Fig. 40 
and 41 answer to these con- 
ditions, for from C to D the 
first differential coefficient is 
negative (Art. 36) and de- 
creasing, while from D to E 
it is positive and increasing in both cases. 

If the second differential coefficient is negative, then the 
first must be positive and decreasing, or negative and in- 
creasing, and we find the curves in Fig. 39 and 42 to answer 
these conditions ; for from C to D the first differential coeffi- 
cient (Art. 36) is positive and decreasing, while from D to 
E it is negative and increasing in both cases. 

By inspecting these figures we see that for 39 and 40 the 




SECOND DIFFERENTIAL COEFFICIENT. 189 

second differential coefficient has in each case a sign con- 
trary to that of the ordinate, and that both curves are con- 
cave to the axis AB ; while in curves 41 and 42 the sign is 
the same as that of the ordinate, and the curves convex to 
the axis. Hence 

When the signs of the second differential coefficient and of the 
ordi?iate are contrary to each other, the curve will be concave 
toward the axis of abscissas; when these signs are alike the curve 
ivill be convex toward that axis. 

It will be noticed that in all these cases the first differen- 
tial coefficient changes its sign at D where it becomes zero, 
but this does affect the sign nor the value of the second 
differential, for the first may be changing as rapidly, and in 
either direction at the zero point as at any other. 

(94) To illustrate these rules let us take the general 
equation of the circle 

{x-a) 2 +(y-b) 2 =R 2 
in which a is the abscissa and b the ordinate of the center. 

Differentiating we have 



and 



dx 2 {y-b) 3 Fig. 43. 

From which we learn that so long as y is greater than b 
the second differential coefficient will be negative, while it 
is positive where y is less than b, or where it is negative. 
We see also from the figure (Fig. 43) that above the line DE 
where y is greater than b the curve is concave toward the 
axis of abscissas, while between DE and the axis of abscis- 
sas, where y is positive and less than b, the curve is convex 
toward that axis. Below the axis of abscissas where y is 
negative the second differential is still positive, while the 



dy 


x—a 


dx 


y — b 


d 2 y 


R 2 . 



*©!-* 



190 DIFFERENTIAL CALCULUS. 

curve is concave toward the axis. All of which corresponds 
with the rule. 

In the case of the parabola referred to its vertex and 
axis we have 

d 2 y p 2 

dx 2 y s 

a fraction whose sign is always contrary to that of y; hence 
the curve is always concave towards the axis of abscissas. 
The same may be said of the ellipse referred to its center, 
and axes from whose equation we have 
d 2 y B 4 

dx 2 A 2 y 3 

In the case of the hyperbola referred to its center and 
asymptotes we have 

d 2 y zy 
dx 2 x 2 

a fraction whose sign" is always the same as that of y. 
Hence the curve is everywhere convex toward the axis. 

VALUE OF THE SECOND DIFFERENTIAL COEFFICIENT. 

(95) The curvature of a curve at any point is the ten- 
dency of the tangent line at that point to change its direc- 
tion, as the point of tangency is moving along the curve, in 
obedience to the law of change derived from the conditions 
which govern the movement of the generating point. 

Note. — The curvature then of a curve is not " its deviation from the tangent,' 1 * nor 
u its departure from the tangent drawn to the curve at that point, "t nor is it "the 
angular space between the curve and its tangent,":}: nor is it any actual change in the 
direction of the tangent line as the point of tangency moves along the curve ; nor does 
it depend on any such change, but upon the law which governs the movement of the 
generating point ; for it is this law which fixes the tendency of the tangent to change 
its direction and this tendency is the curvature. Hence in estimating the curvature of 
a curve at any point, we consider that point alone and seek, not any actual movement 
of the generating point, but the lazu which controls it. 

*Loomis. tDavies. ^Church. 



V 



SECOND DIFFERENTIAL COEFFICIENT. I91 

Hence if several curves as CD, CD', CD'' (Fig. 44) have 
coincident tangents AB at the point A, and if we suppose 
the point of tangency to be at any instant moving along the 
curve, carrying with it its own tangent ^ a g 

line, that one whose tangent line at ~ ^ 
the moment of coincidence is chang- / / / 
ing its direction most rapidly will G q' c 
have the greatest curvature at that 
point. For the rate of change in the Fi s- 44- 

direction of the tangent is the measure of its tendency to 
change. 

Since the first differential coefficient indicates the direc- 
tion of the tangent to a curve, by means of the tangent of 
the angle made by it with the axis of abscissas ;' the second, 
which is simply the rate of change in the first, will indicate 
the rate at which the tangent of that angle is changing its 
value. Now as between two curves at common tangent 
point, that curve in which the tangent line tends to change 
its direction most rapidly, will be the one in which the 
tangent of the angle made by that line with the axis of 
abscissas will also tend to change its value most rapidly, and 
will, therefore, have the greatest curvature, while if these 
tendencies are equal the curvatures are equal, and this 
will be indicated by the equality of the second differential 
coefficients. 



SECTION XII. 



CURVATURE OF LINES. 



THEOREM. 



(96) The curvatures of different circles are inversely propor- 
tional to their radii. 

The curvature of a circle is the same at all points of the 
circumference, and all circles having the same radii have 
the same curvature. 

Since the change in the direction of the tangent, as the 
point of tangency moves around the curve is constant, its 
actual change of direction for any given movement of the 
point of tangency, will always be in proportion to its ten- 
dency to change, multiplied by the length of the arc over 
which the movement is made, and may, therefore, be repre- 
sented by that product ; and hence the tendency to change or 
curvature will be equal to the actual change divided by the 
length of the arc. 

Now the change in the direction of the tangent is equal 
to the angle contained between its two positions, which is 
the same as that contained between the two radii drawn to 
the extremities of the arc. Calling this angle v and the 
length of the arc #, we shall have 

v 

curvature = — 

a 

192 



CURVATURE OF LINES. 



193 



If now we have two circles, which we will call o and d , 
whose radii are r and r\ and the angles at the center for the 
same length of arc a are v and z/, we shall have 

curvature of <?=— 



hence 

but 

and 

whence 

or 



curvature of d=— 
a 



curvature of : curvature of d : : v : %f (1) 

v\ 360 : : a 1 2-xr 
v r : 360 : : a : 21:?-' 



v o 2~T—V . 2~/ 



v\v \\r tr 



Substituting this ratio in proportion (1) we have 
curvature of : curvature of d \\r \r 



Q. E. D. 



CONTACT OF CURVES. 



(97) When two curves have a common point, the coordi- 
nates of that point must satisfy both their equations. This 
will generally be a point of intersection, and not a point of 
contact ; and is all that can be secured by having but one 
condition common to the two curves. 

If they are at the same time tangent to each other, at the 
common point, then another common condition is imposed 
and there is a contact of the first order. 

The condition required in this case is, that, for the point 
of contact, the first differential coefficients shall be the same 
for the equations of both curves. For since the curves are 
tangent to each other, they have a common tangent line, and 



IC)4 DIFFERENTIAL CALCULUS. 

the first differential coefficient, which determines the angle 
made by this line with the axis of abscissas, must be the 
same for both equations. 

If, besides this, the curves are required to have the same 
curvature at the point of contact, this will introduce a third 
condition, which is, that the second differential coefficients 
shall be the same for both equations (Art. 95). 

For the second differential is the rate of change in the 
first, which gives the direction of the tangent line, and the 
rate. of change in this direction is the curvature. This is a 
contact of the second order. 

If now it is required, in addition, that the rate of change 
in the curvature should be the same in both curves at the 
point of contact ; we must introduce a fourth condition, 
viz., that the third differential coefficient should be the same 
in both equations. This would be a contact of the third 
order. And thus the order of contact would become higher 
for every new condition introduced common to both curves, 
and every new agreement between the successive differential 
coefficients. 

If then we wish to find the order of contact of two given 
curves, we first combine their equations, and determine their 
common point if they have one. For this point the varia- 
bles will have the same value in both equations. If the 
values thus found being substituted in the first differential 
coefficient of each equation, reduce them to the same value, 
there is a contact of the first order; that is, they have a 
common tangent line at the common point. 

If they also reduce the second differential coefficients of 
the two equations to the same value they have a contact of 
the second order, and so on for the successive differential 
coefficients ; the order of contact being determined by the 
number of coefficients that successively become equal by 
the substitution of the values of the common coordinates. 



CURVATURE OF LINES. 1 95 

EXAMPLE. 

(98) To illustrate this rule let us take the two equations 

-ti'=^-4 (1) 

and 

y 2 — -_J' = 3 — x 2 (2^ 

from which we obtain by combination 

_r = — 1 and x=o 
indicating that both the curves pass through the point of 
which these are the coordinates. We have also by differen- 
tiating twice — for equation (1) 

dy__x d 2 j' _ i 

dx ~~ 2 dx 2 ~ 2 

and for equation (2) 



dy x d 2 v 

and 



dx y—\ dx 2 J— 1 (/— i) 3 

Substituting in these differential coefficients the values of 

x and v just found, we have the first differential coefficients 

dv x dy x 

=0 and 



dx 2 dx y — 1 

and the second differential coefficients 

d 2 y d 2 v 1 

-~i and 



dx 2 -2 «•"" dx 2 y _ 1 ( 7 -j)3 2 

from which we infer that at the point whose coordinates are 
.v = o and j ,=z — 1, the curves have a contact of the second 
order. We also see from the value of the first differential 
coefficient that at that point the tangent to both curves is 
parallel to the axis of abscissas. A little investigation 
would show that the first curve is a parabola, and the sec- 
ond a circle tangent to the first at its vertex. 

(99) The constants which enter into the equation of a 
curve determine the conditions which govern the movement 
of the generating point for that kind of curve ; which must 
fulfil as many conditions as it has constants. Thus the cir- 



I96 DIFFERENTIAL CALCULUS. 

cle whose general equation contains three constants, must 
fulfil three conditions, namely, two in the coordinates of the 
center, and one in the length of the radius. The ellipse 
must fulfil four conditions, namely, the coordinates of the 
center and the lengths of the two axes. 

(100) Now if one curve be given complete by its equation 
with fixed values for its constants, and another with con- 
stants which are indeterminate, and capable of being adjusted 
to any given conditions, we may easily assign such values to 
them as will cause the curve to fulfil such conditions as 
may be required of it. We may, for instance, require the 
curve to pass through a given point in a given curve. This 
will require that the same variable coordinates shall satisfy 
the equations of both curves for that point. "We may also 
require them to have a common tangent at that point ; this 
Avill require the constants to be so adjusted that the first 
differential coefficients of the two equations shall be equal. 
If there are three or more constants in each equation we may 
require such values as will cause the second differential 
coefficients to become equal also, thus producing an equality 
of curvature, or a contact of the second order, at the com- 
mon point. And thus we may continue until the order of 
contact is one less than the number of constants to be dis- 
posed of. 

(101) In order to make this adaptation of the second curve 
to the first we must consider its constants, or as many of 
them as will be required for the purpose as unknown quan- 
tities (Art. 4) and construct as many equations as may be 
required to determine them. 

These equations are derived from the conditions to be 
fulfilled by the constants. Thus the first which requires 
that the second curve shall pass through a point of the first 
will generally be met by the proper adjustment of a single 
constant ; and an equation formed by substituting in that of 



CURVATURE OF LINES. 1 97 

the curve to be adjusted the values of the coordinates of the 
designated point, and also the values of the known con- 
stants, will determine the value of the unknown constant. 

If it is required that the two curves be tangent to each 
other, we must adapt the values of two constants to this 
condition, and this is done by substituting the same values 
of the common coordinates, and of the remaining constants 
in the first differential coefficients of the two equations, and 
placing them equal to each other, thus forming a second 
equation. A contact of the second order may be secured 
by fixing the value of a third constant in a similar way by 
means of the second differential coefficients of the two equa- 
tions. 

The values of these constants thus determined bei?ig substituted 
in the general equation of the required curve, will produce an 
equation of one that will fulfill the required conditions; that is, 
one that will intersect at a given point, or have a contact of 
a required order. 

EXAMPLE. 

(102) To illustrate these principles let us take the equa- 
tion of the ellipse referred to its center and axis 
Ay+B-«*»=A»B a 

and the general equation of the circle 

(x-a)2+{y-b) 2 =R* (i) 

in which the constants are arbitrary and may be adapted to 
any prescribed conditions. Suppose we say that the cir- 
cumference shall pass through the upper extremity of the 
conjugate axis where 

x=o and y=B 
This being but one condition will require the adaptation 
of but one constant. Let this be a, while we make R— A 
and 0=0. 



198 DIFFERENTIAL CALCULUS. 

Then substituting these values in equation (1) we have 

07-tf) 3 +(B-<7) 2 =A 2 

or 

<Z 2 +B 2 =A 2 

whence 

a=±VA 2 -B 2 
and the equation of the circle becomes 

(^TVA^-B 2 ) 2 +y 2 =A 2 
the center being in one of the foci — the plus value of the 
radical corresponding with the focus on the positive side of 
the center. 

If we add another condition, namely, that the curves shall 
be tangent to each other at the same point, we must adapt 
the value of two constants to these two conditions. Let 
these constants be a and ^, and make R = 2B. Then we 
must construct an equation between the first differential 
coefficients of the curves ; that is 

B 3 .t x—a 

Uy^ y-b ' 2 ) 

Substituting the values of x and y as before we have 

B 2 o _ 0— a 

A^B^B-^ 
hence 

a=o 
and substituting these values in equation (1), we have 

(-B-o) 2 = 4 B 2 
whence 

b = -B 
and the equation of the required circle becomes 

x 2 +(y+B) 2 =4~B 2 
the center being at the lower extremity of the conjugate 
axis where a=o and ^ = — B. 

If now we add a still further condition there shall be a 
contact of the second order at the same point we must adapt 



CURVATURE OF LINES. 199 

the values of three constants to that condition, by forming a 
third equation, between the second differential coefficients, 
thus 

df (x-a)* 

B 4 _ 1+ dx* __ 1+ {y-bY 
A 2 /" y-b ~~ y-b ^ 

Substituting, as before, the values of x—o and y—B in 
equations (1), (2), (3), we have three equations from which 
to determine the values of the three constants ; thus 
(o-ay+{B-b) 2 =R* 
B 2 o o—a 







A 2 B~ 


~B- 


-b 
-a) 2 






B 4 x 


'(B 


-by 






A 2 B 3 ~ 


B- 


-b 


From 


the second " 


we obtain 

a- 


—0 




From the third we 


\ have 










B 3 - 

h— 


A 3 



B 

and substituting these values in the first we obtain 

A 3 

and the equation of the circle becomes 

. . B 3 -AV A 4 

the radius being equal to half the parameter of the conju- 
gate axis of the ellipse, and the center being in that axis 
prolonged in a negative direction. 

(103) In this last case we have the highest order of con- 
tact of which the circle is capable, and hence the circle is 
called the osculatrix to the ellipse; or is said to be oscula- 
tory to it. 

An oscidatrix to a curve is one which has the highest order of 



200 DIFFERENTIAL CALCULUS. 

contact with it, that any curve of the same kind as the osculatrix 
can have. 

Since the number of constants limits the number of con- 
ditions that can be assigned to a curve, and since the pass- 
ing of the curves through the same point is one condition, 
the order of contact can only be equal to the remaining 
number of possible conditions ; namely, the number of con- 
stants, less one, which enter into the general equation ; and 
this will be the same as the order of its highest differential. 

EXAMPLES. 

(104) Ex. i. To find the equation of the circle oscula- 
tory to the parabola, whose equation is . 

y 2 =4X (i) 

at the point where the coordinates are 
x = i and y =z 2. 
Differentiating this equation we have 



whence 



and 



or 



7 — ano. 7 s> — — — o 
ax y ax* y 6 



2 x — a 

y~~y—b 

y 3 y — b 

i— a 



and 



2-b 



(\—a\ 



2 



1 - 

2—b 



(2) 



(3) 



CURVATURE OF LINES. 201 

Also from the general equation of the circle we have 

(l-tf) 8 +(2— £) 2 =R 3 (i) 

and from these we. find 

R 3 = 3 2, 0=5, £ = — 2 

and the equation of the circle oscillatory to the parabola at 
the given point is 

(^-5) 3 +(^ + 2) 3 = 3 2 

Ex. 2. To find the circle osculatory to an equilateral 
hyperbola whose equation is 

xy=8 
at a point whose coordinates are 

y—\ and x=2. 
By differentiating we have 



and 



and from the general equation of the circle we have 

(2-ay+(4-&y=R* (i) 



dy = _ 
dx 


X 


— 2 


d 2 y 


2y 




dx 2 


~ x 2 ~ 


-2 



' 7 — 2 



(2—CLY 



7 ~^ — 2 



from which we obtain 



R 3 =— ^=7 £=— 

4 ' 2 



(2) 
(3) 



and the equation of the required circle will be 

<*-7>'+(,-^)'=f 

Ex. 3. Find the equation of the circle osculatory to the 
curve whose equation is 



DIFFERENTIAL CALCULUS. 



4y=x 2 — 4 



at a point whose coordinates are 

X=-o y= — i 

RADIUS OF CURVATURE. 

(105) Since the curvature of a curve at any point is the 
same as that of its osculatory circle at that point, we call 
the radius of the osculatory circle the radius of curvature of 
the curve. And since the formulas for the equation of the 
osculatory circle may be applied to any point of a given 
curve, we may consider them as expressing the general con- 
ditions required of the osculatory circle. 
These formulas, as we have seen, are 

(x-a) 2 +(y-b) 2 =R 2 (i) 

dy _ x—a 

dx y—b ^ ' 

dy 2 



d 2 y dx' 

dx 2 y—b 

the two last may be written 

dy 
x — #— — ~- 

and 



(3) 



a= ~dx^-^ ^ 



dx 2 +dy 2 

y-*=- d * y (3) 

If we represent the coordinates of any given point in a 
curve by x and j/, then for the osculatory circle we must 

have 

, f dy dy d 2 y d 2 y 

-> j J *> dx dx'' dx 2 dx 2 
The quantities a and b represent the coordinates of the 
center of the osculatory circle, and R is its radius. 

If we substitute in equation (2) the value of y—b, we 
have 



CURVATURE OF LINES. 203 

dy (dx 2 -\-dy 2 ^ 



x—a=~ 



- 



) 



dx\ d 2 y 

whence equation (i) becomes 

dy 2 /dx 2 +dy 2 \ 2 / dx 2 + dy 2 \ 2 _ 
dx*\ d 2 y ) ~K d 2 y ) ~~ R 

from which we have 

{dx 2 +dy 2 f 2 
^~ ± dxd 2 y ^ 5) 

which is the general expression for the value of the radius of 
curvature in terms of quantities belonging to a given curve. 
If we denote the length of the curve by u we shall have 

du 3 
dxd 2 y 

(106) Since the curve and its osculatory circle have a 
common tangent, they will also have a common normal ; 
and as the normal to the circle passes through the center, 
the normal to any curve at any point will pass through the 
center of the circle osculatory to it at that point. 

This is also shown from equation (2) which is 

dy _ x— a 

dx y — b 

x and y being coordinates both to the given curve and to 

the osculatory circle at the point of contact, and a and b the 

coordinates of the center of the circle. 

dy . 
For since -7— is the tangent of the angle made by the 

tangent line with the axis of abscissas, we shall have 

dx y—b 

dy x — a 
for the tangent of the angle made by the normal line with 
the same axis. But when a straight line passes through two 
points — x and y being the coordinates of one, and a and b 
the coordinates of the other — the tangent of the angle 



204 DIFFERENTIAL CALCULUS. 

made by that line with the axis of abscissas will be 

y—b 
_ , and hence the normal to the curve, since it passes 

through the first point will also pass through the second — 
that is, the center of the osculatory circle. 
And since from equation (3) we have 

d 2 y dy 2y 



jol y ay \ 



the value of the first member of the equation, will be essentially 

d 2 y 
negative, and hence we infer that y—b and , 2 must have 

d 2 y . 
contrary signs. So that if , ^ is negative, b will be less 

than y, and, if positive, it will be greater. In the first case 
the curve will be concave toward the axis of abscissas, and 
b will be between the curve and that axis ; while in the 
other case the curve will be convex toward the axis of 
abscissas, and b will be beyond it. Hence the center of the 
osculatory circle will be on the concave side of the curve. 
(107) To find the general expression for the radius of 
curvature of the parabola, we differentiate its equation twice 
and obtain 

ydy^fidx 
and 

whence 



yd 2 y-\-dy 2 =o 
pdx 



dy- 



and 



y 



dy 2 p 2 dx ? " 
d 2 y = — ^-=-^-3- 



Substituting these values in the formula we have 
3. 

f> 2 dx 2 \ 2 

_ ^ 2 +-^ J [, /a .8( J ,8 +J> g)]f _ ( 2/ , y+ ^g)t 



yZ 



CURVATURE OF LINES. 205 

or, the cube of the normal (Art. 56) divided by the square 
of half the parameter. 

If we make x—o we have 

or half the parameter for the radius of curvature at the 
vertex. If we make x=-\p we have 

for the radius of curvature at the point where the ordinate 
through the focus meets the curve. As every other value 
of R is greater than that where x'—o it follows that the 
greatest curvature of the parabola is at the vertex. 

(108) From the equation of the circle we have 

xdx 

dy : = — ■ 

J 

and 

R 2 dx 2 

and substituting these values in the formula we have 



x 2 dx 2 \ 



3 

8 



x 2 dx 2 \ 2 
W~' _[{y 2 +x 2 )dx 2 ] 2 



R -R 2 dx« -RV* 3 ~~ R 

y 

the radius of the circle as it should be. 

(109) From the equation of the ellipse we have 

~B 2 xdx 

and 

or substituting in the last equation the value of dy 2 we have 

B±dx 2 

These values being substituted in the formula 



2o6 DIFFERENTIAL CALCULUS. 



R 



B±x 2 dx 2 \ 2 /A 4 y 2 +B 4 * 2 \ 8 

_ BV.r 3 B* 

A 2 y "A 1 



which is equal to the cube of the normal divided by the 
square of half the parameter as in the parabola. 
If we make #=A we have j=o and 

R — A 
If _y=B then x=o, and we have 

A 3 

R= -F 

Hence the radius of curvature of the ellipse at the princi- 
pal vertex is half the parameter o f the transverse axis — 
that is the ordinate through the focus. At the vertex of the 
conjugate axis, the radius is half the parameter of that axis 
(Art. 102). 

(110) The equation of the hyperbola referred to its center 
and asymptotes gives 

ydx 

dy= — ■ 

J x 

and 

2dxdy 

d 2 y=- — ' 

J x 

Substituting these values in the formula we have after 
reducing. 

_ (x 2 +y 2 ) 2 _ 2(x 2 +y 2 ) 2 
K ^~ 2xy ~~ A 2 +B 2 
In the equilateral hyperbola, this value becomes equal to 
the cube of the radius vector divided by the square of the 
semi-axis. 



SECTION XIII. 



E VOLUTES. 

(111) If we suppose a circle to roll along the concave 

side of a curve, being always tangent to it, and at the same 
time varying the length of its radius so as to be osculatory 
also, its center will describe a curve which is called the 
evolute of the given curve ; and its variables will be the 
coordinates of that variable center. In other words, the 
e vol Lite of any curve is the locus of the centers of all the 
circles that can be drawn osculatory to that curve. 

The relation between the variables of the evolute can be 
determined and its equation found from the equation of the 
given curve, and the first and second differential coefficients 
derived from that equation ; since these determine the posi- 
tion and length of the radius of curvature, and consequently 
the place of the center of the osculatory circle. 

Since the coordinates of the point of tangency and the 
first and second differential coefficients are the same for the 
given curve and for the osculatory circle, we can at once 
determine two of the properties of the evolute. 

(112) The first of these properties is, the radius of the 
osculatory circle is tangent to the [C 
evolute. * 

Let AC (Fig, 45) be any curve, 
and let c be the center of the oscula- 
tory circle for the point A, while c , 
c\ c" r are the centers of the oscula- 
tory circles corresponding to the 

207 




2 0cS DIFFERENTIAL CALCULUS. 

points A', A', A'". Then the curve cc' n passing through these 
centers will be the evolute, and any radius as AVwill be tan- 
gent to it at the point c\ the center of the osculatory circle. 
The equations of conditions (Art. 105) may be put into 
the following form 

(x-a) 2 +(y-b) 2 =R 2 (1) 

(x — a)dx + (y—b)dy—o (2) 

(y-b)d 2 y+dy 2 +dx 2 = o (3) 

and in this case a, b, R, x, y are variables ; x being indepen- 
dent and dx a constant quantity ; while x and y are coor- 
dinates of the given curve, and of the osculatory circle at 
the point of contact, and a and b coordinates of the varia- 
ble center of the osculatory circle, that is, of the evolute, 
and are functions of x and_y. 

From these equations, as we have seen (Art. 105), R may 
be determined for any point in the given curve by eliminating 
a and b considered as constants. But for the evolute curve 
we must consider them as variable coordinates ; and hence 
under that supposition if we differentiate equations (1) and 
(2) we have 

(x—a)dx + (y—b)dy—(x—a)da — (y—b)db=RdR (4) 
and 

dx 2 +dy 2 -\-(y— b)d 2 y — da . dx— db . dy—o (5) 

Subtracting equation (2) from (4) we have 

-(x-a)da-(y-b)db-RdR (6) 

and subtracting (3) from (5) we have 

— da . dx — db . dy = o (7) 

whence 

db dx 

~da~^~~dy (8) 

dx . 
but — — — is the tangent of the angle made by the normal 

line to the curve, at the point whose coordinates are x and y, 

db 
with the axis of abscissas * and —7- the tangent of the an- 



EV0LUTE9. 209 

gle made by the tangent line to the curve at the point whose 
coordinates are a and b with the same axis. But x and y 
are coordinates of the given curve, and a and b are coordi- 
nates of the evolute, and, of course, of the center of the 
oscillatory corresponding to the point (x .y) on the curve, 
and through this center the normal line must pass (Art. 
1 06) ; and since both the normal to the curve (or radius of 
curvature) and the tangent to the evolute pass through the 
same point, and make the same angle with the axis of 
abscissas, they must be one and the same line ; and hence 
the proposition. 

(113) The other property referred to in Art. 1 1 1 is 
The difference between the length of the evolute curve and the 
radius of curvature, measured from the sa?ne point is either zero 
or a constant quantity. 

From equations (2) and (8), of the preceding article, we 
have 

da 

x - a =~db\y- h ) (9) 

and substituting this value of x— a in equation (1) we have 
x <to* , x . , , da 2 +db 2 , , 

(■y-^) 2 ^i+Cy-^) 8 = R8 =(^-^) 8 a* (lo) 

From equation (9) and (6) we have 

da* x , ' da?+db* t 

which being squared gives 

N (da? + d&*)* 

and this being divided by equation (10) gives 

da 2 +db 2 =dR* 
If we designate the length of the evolute by u we shall have 

du*=da?+db* 
whence 

du 2 =dR 2 



2IO DIFFERENTIAL CALCULUS. 

or 

du=dR or d~R— du=o=d(R— u) 
hence R— u is a constant quantity and 

R=u+c 
If 71=0 we have 

and hence c is equal to the radius of curvature at the begin- 
ning of the curve, and R is at all times equal to the length 
of the evolute to the point where R is tangent plus the con- 
stant c. 

If, therefore, we suppose a cord to be fastened at B (Fig. 
45) and drawn tight around the curve AB and then unwound 
from A, the end of the cord will describe the curve AC of 
which the curve AB is the evolute. For the cord will be at 
all times tangent to the curve from which if is unwound, and 
also the momentary radius of the curve AC for the point at 
its own extremity, and consequently normal to the curve at 
that point ; while the length of the cord from the point of 
tangency to its extremity in the curve AC is equal to the 
distance from the same point to the origin at A measured 
along the curve AB. 

(114) To find the equation of the evolute, we must com- 
bine the equation of the osculatory circle with that of the 
involute in such a manner that x .y and R shall disappear 
and leave an equation containing only a and b as variables. 

This will require four equations, and these are obtained 
from the equation of the involute, the general equation of 
the circle, and those formed by placing the first and second 
differential coefficients of each of these equations respec- 
tively equal. 

Thus if we take the equations of condition (Art. 105) 

{x-a) 2 +(y-b) 2 =R* (1) 

dy dy x — a 

x ~ a= -^y-^ or &=--;=* (2) 



EVOLUTES. 211 

df_ 

dx 2 + dy 2 d 2 y _ * + dx 2 
y d % y dx 2 y — b ™' 

and then differentiating the equation of the involute twice? 
we find the values of the same differential coefficients and 
make them equal to the second members of equations (2) 
(3) ; then eliminate x,y and R, the resulting equation is that 
of the evolute. 

Since R is contained in only one equation, we omit that, 
as the remaining three are sufficient for eliminating x and 7, 
and for the resulting equation. 

(115) To find the equation of the evolute to the parabola. 

The equation of the parabola is 

y2=z 2 pX (i) 

from which 

dx y ^ ' 

and 

dx 2 y 3 ™' 

Placing these differential coefficients equal to those of the 
general equation of the circle, we have 
/_ x— a 

(4) 

and 



y 


y—b 


p 2 


df~ 

1 ' dx 2 


y*~ 


- y-b 



(5) 

Dividing equation (5) by equation (4), and substituting for 
y 2 its value from equation (1), and reducing, we have 

a = ^x+p (6) 

and substituting the values of a and y in equation (4) we 
have after reducing 



*=- 



(ix)^ 



p 2 



DIFFERENTIAL CALCULUS. 



and substituting in this the value of x from equation (6), 
and squaring, we have 



£2=- 



\a-pY 



= .T->-/) 2 



*B^ 



3 3 / 2 lP x 

which is the equation of the evolute of the parabola. 

If we make b=-o we have a=-fi, which is the center of the 
osculatory circle for the vertex. If we transfer the origin to 
that point we have 

a=j>+a f and b=b r 
hence 



b' 2 = — a'* 

27 

Since every value of a gives two 
equal values for b' with contrary 
signs, the curve of the evolute ACE 
(Fig. 46) is symmetrical about the 
axis of abscissas. If a is negative 
then b f is imaginary, and hence the 
curve commences at C, a point in the 
axis of abscissas at a distance from 
A equal to/ — that is, at double the 
distance of the focus, or half the parameter. 

(116) To find the equation of the evolute of the 
For this case we have 

Ay+BV=A 2 B 2 
dy B 2 x x — a 

dx A 2 y~ y — b 

dy 2 
dx- 




d 2 y 
dx 2 , 



B 4 



1 + ^„2 



A 2 y 



2 1; 3 



y 



-b 



ellipse. 

(1) 
w 

(3) 



From equations (2) and (3) we have 



y—b= 



A 2 y(x—a) 
B 2 x : 



AVd-t^r) 



B 4 



EVOLUTES. 213 

x-a_ y ^^ dx 2 > _ y ^ I + AV 3 ' 
x ' B 2 """" B 3 

B 2 (*-tf) = 4 ji ) 

A±B 2 x—A±B 2 a=A±xy 2 +B±x s 

A 2 x(A 2 B 2 -A 2 y 2 )=A±B 2 a+B 4 <x 3 

A 2 B 2 ^ 3 =A 4 B 2 tf+B% 3 

a = A 4 xS (4) 

Substituting this value of a in equation (2) we have 
A 2 -B 2 _(7-^)B 2 ^ 



whence 

whence 

whence 
whence 
whence 
whence 



A ^ "* -- A 3 y 

whence 

A 4 -(A 2 -B 2 >t: 2 _ (j-l)B 2 

A 2 "" j 

whence 

A 4 j;-AVj+BVj=A 2 B 3 j/-A 2 B^ 
whence 

XA 2 -* 2 -7 2 ) = -B 2 £ 
Substituting for .t 2 its value from equation (1) we have 
, o A 2 B 2 -A 2 y 2 

whence 

A s -B 3 , 

*=— Bi-y s (5) 

Making A 2 — B 2 =c 2 we have 

c 2 c 2 

a ~~A± x3 anc * b=— ~^y 3 



214 DIFFERENTIAL CALCULUS. 



and making ~j~—m and -rr=77, we have 
a x 3 b y 3 

771 A 3 71 B 3 

or 

_i i 

, 3 , y rb\ 3 



x f a\° y (fry 

"X = W and B" = -U 



Writing the equation of the ellipse under the form 

x 2 v 2 
A* + B^ 



2+-D2— X 



x ■ y 
and substituting the values of ~r~ and -77- just found, we 

have 

2. 2 

O'+OS*^ ( 6 ) 

which is the equation of the evolute of the ellipse in which 
a and b are the variable coordinates, and m and n the con- 
stants. If we make a= z o we have b~±n, and if ^=o we 
have <2=±//z, which shows that the form of the evolute is 
symmetrical with both axes of the ellipse. But 

C 2 a 2 -B 2 

and subtracting this from A we have the radius of curvature 
at the principal vertex equal to — r~, as we have already seen 

(Art. 109). Similarly we find the radius of curvature at the 

A 2 
vertex of the conjugate axis to be ~fT. 



If we differentiate equation (6) twice we have 

1 / a\ 3 1 / b\ 3 db 
771^771' 7i\ 71' aa 



whence 



EVOLUTES. 



215 



and 



db_ 
da 



-A 
3 






3 






m f an\ 3 

n^ bm' 



_4. 



i / a\ ° i / u\ °db* 3[ v\ 
m 2 \m) n 2 V n' da 2 /A n> 



~*d 2 



da 2 



whence 



b\ 3 db 2 



d 2 b 



i r a\ ° i / (?\ °db* 
n 2 \m' n 2 V n> da 2 



da 2 



i 1 ) 




Since the numerator of the second differential coefficient 
is always positive, it will have the 
same sign as the denominator, which 
is the same as that of <£, and hence A i 
the curve is everywhere curvex 
toward the axis of abscissas. The 
first differential coefficient becomes 
zero when a=o, and infinite when b=o, hence both axes 
are tangent to the curve, as in Fig. 47. 

If we make A=B, then £=<?, and also m=o and n=a, 
hence a and b in equation (7) will also become zero as they 
should, since in case of the circle the evolute is reduced to 
a point — the center. 



Fig. 47- 



SECTION XIV. 



ENVELOPES. 



(117) Suppose two lines, AB and AC (Fig. 48), be drawn 
at right angles to each other, and a third line ed to move in 
such a manner that its extremities d and e shall be con- 
stantly in these axes, while its length remains unchanged; 
so that while the extremity e arrives 
successively at the points e\ e\ the 
extremity d will arrive at the corres- 
ponding points d\ d" . 

During this movement those points 
of the line near the extremity d will 
move in the direction more nearly 
parallel to the axis AC than the line 
itself is, and will consequently fall within its first position, 
while the points near the extremity e will move in a direc- 
tion more nearly parallel to the axis AB than the line is, and 
will consequently fall without its first position. But between 
these extreme points there is one that tends to move in the 
direction of the line itself. 

This point does not, of course, remain fixed on the line, 
but moves from one extremity to the other as the line changes 
its position and direction, always occupying that place in the 
line which at the moment does not tend to move out of it 

216 




ENVELOPES. 



217 





E 


c" 




M 


£L 


^ 


^B 


^K 


^ 




y 




V 


/ 





towards either side. 77^ ^/w described by this point is the 
envelope of the curve. 

Again let AB (Fig. 49) be the transverse axis of an 
ellipse, and CD its conjugate axis ; and suppose these axes 
to vary to any extent under the condition that the area of 
the ellipse shall remain constant. 

Then as AB decreases CD will 
increase at a rate corresponding with 
this condition. When the curve 
thus commences to change its shape, 
a point near the extremity A will 
tend to move in a direction more 
nearly parallel to AB than the tan- 
gent to the curve at that point is ; 
while a point near the extremity C Fig. 49 . 

will tend to move in a direction more nearly parallel with 
CD than the corresponding tangent line is. Now between 
A and C there is a point in the curve that tends (as the 
axes are changing) to move exactly in the direction of the 
tangent to the ellipse at that point. 

As the curve changes its shape and position this point 
will also change its place on the ellipse, keeping always 
where its tendency is in the direction of the tangent to the 
ellipse as it is at the moment. The movement of the point 
will be continuous, and it will generate a curve which will 
be the envelope of the ellipse, 

(118) Since the point on the given curve which describes 
the envelope always tends to move in the direction of the 
momentary position of the tangent to the curve at that 
point, and since any generating point always tends to move 
in the direction of the tangent to its own curve, it follows 
that the given curve and its envelope will have a common 
tangent line wherever the generating point may be at "the 
moment during the formation of the curve. Thus in the 



2l8 DIFFERENTIAL CALCULUS. 

last illustration, the ellipse, in every stage of its change, will 
be tangent to the envelope at that point of the curve just 
then generated. 

(119) An envelope to any line, is another line generated by that 
point of the given line, which tends to move in the direction of the 
tangent, whenever its position or shape is made to cha?ige by chang- 
ing the constants of its equation, or any of them, into variables. 

An envelope is not always produced by this change of the 
constants, for it may be that no point of the given line will 
tend to move in the direction of its tangent; as in the case 
of an ellipse where both axes are increased. 

In general, there will be an envelope only where the suc- 
cessive positions of the line corresponding with minute 
changes in the constants, will intersect each other; for while 
the generating point of the envelope tends to move in the 
direction of the tangent, the points on each side of it will 
tend to move away from the tangent in opposite directions, 
hence the next position of the changing line will cross the 
previous one near the generating point of the envelope. 

(120) If in any equation of a line the constants are made 
to vary in value, it is evident that while the curve or line 
remains the same in kind, its shape and position may assume 
every possible form and place within the limits determined 
by the law of variation imposed upon the constants of the 
equation. 

If we take for example the ellipse, and consider A and B 
in its equation as independent variables, then 

A^ 3 +B 2 ^ 2 =A 2 B 3 
will represent an infinite number of ellipses of every possi- 
ble size and proportions subject to but two conditions; 
namely, the axes must both coincide with the axes of coordi- 
nates. If we make A and B dependent on each other we 
limit the system of ellipses by the condition thus introduced, 
but still their number is infinite. If we introduce the still 



ENVELOPES. 219 

further condition that the values of x and y shall be confined 
to those points of the system which tend to move in the 
direction of the tangent, while A and B tend to change their 
values, the first differential coefficient will not be affected by 
such tendency in A and B, and hence will be the same at 
those points whether they are considered as variables or con- 
stants. So then if we take the differential of the equation 
with respect to them only as variables, and make it equal 
to zero, and incorporate it with the original equation, we 
put this limit on the values of x and j, which will then 
only apply to points in the envelope. The equation will, 
therefore, be that of the envelope itself — that is, instead of 
representing every point in one ellipse, it will represent one point 
in each quadrant of every ellipse that tan be formed under the 
given conditions. 

To find the equation then of an envelope we differentiate 
the equation of the given line with reference to such only 
of the constants as are considered variable for the time 
being, and place that differential equal to zero. The values 
of the constants determined from this equation, and the 
conditions of relation among themselves, being substituted 
in the given equation, will produce one that will be inde- 
pendent of the variable constants, and this will be the equa- 
tion of the envelope. 

EXAMPLES. 

(121) For the first example, let us take the general equa- 
tion of the circle in which R and b are constants, while a is 
considered as a variable. Now since the values of x and y 
are to be confined to those points of the circle which tend 
to move in the direction of the tangent while a varies, it 
will make no difference whether we differentiate with re- 
spect to x and y only, or with respect to a also. Differenti- 
ating in both these ways we have 



220 



DIFFERENTIAL CALCULUS. 



and 



(x—a)dx+(y—&)dy=o 



(x—a)dx—(x—d)da-\-(y—b)dy—o 
making these differentials equal, and cancelling like terms 
we have 

— {^—d)da—o (i) 

which we should have obtained at once by differentiating 
with respect to a alone, considering all the rest as constants. 
From equation (i) we have 

x=a 
and this value substituted in the general equation gives 
y— £=±R or y=Z?±R 

If we take the positive value for R, this is the equation 
of a line DE (Fig. 50) 
parallel to the axis of ab- C 
scissas at a distance equal 
to that of the centers of 
the system of circles plus 
the radius, and hence tan- ^ 
gent to them all on the 
upper side, and is genera- 
ted by the highest point of the circle as it moves from D to 
E, as a varies in value ; that point being the one that tends 
to move (and in this case does move) in the direction of the 
tangent to the circle drawn through«it. If we take the neg- 
ative value of R, the equation represents the line D'E' tan- 
gent to the system of circles on the lower side. 

(122) If we take the same equation and consider a and b 
both as variables, we must establish a relation between them 
in order to make them both functions of x and y. Let this 
relation be expressed by the equation 

then the two equations will represent a system of circles 




Fig. 50. 



ENVELOPES. 



221 



(Fig, 51) whose centers lie in the 
circumference of another circle 
whose radius is equal to c, and its 
center is at the origin. 

Differentiating the general equa- 
tion of the circle with respect to a \ 
and b only we have 

db 



— (x— a)—{y- 

whence 



-b) 



da 

db_ 
da 




Fig- 5i. 



x—~ a 
y—b 



We may now substitute for b its value obtained from equa- 
tion (1); or we may consider it as a function of a in that 
equation and substitute the value of the differential coeffi- 
cient derived from it. This will give us 

db _ a _ x-—a 

da" b~ y—b 
whence 

\x—a) — £ 3 

Substituting this value of (x — a) 2 in the general equation of 
the circle, we have 

(c*-b 2 ){y-b) 2 



from which we obtain 



and similarly 



+ (j,_£)8 = K > 8 



cy 



c±R 



ex 



Substituting these values of a and b in the general equation 
of the circle we have 

ex \ 2 cy \ 2 



c±R 



222 DIFFERENTIAL CALCULUS. 

whence 

i»+>»=(*±R)i 

the equation of the envelope showing it to be twofold. The 
positive value of R gives a circle with a radiu.s equal to c-\-~R 
circumscribing the system, and the negative value for R 
gives one that is inscribed within it. 

(123) Let there be an ellipse in which the axes vary in 
length under the condition that the area of the ellipse shall 
be constant. This condition will be expressed by the equa- 
tion 

AB=* S (i) 

To find the envelope of this curve we put its equation 
under the form 

and differentiating with respect to A and B only we have 
x 2 y 2 dB_ 
Aj+B 7 • dA~° 
or 

i x 2 i dB y 2 

X • A^-lf • dA' B*" 
But from equation (i) we obtain 

JL- J_ ^? 
A - "" B * dA 

whence 

x 2 __y 2 __ 

A 2 '~~B 2 ~~~ 2 
whence 

&=x\Z~2 and B^j/yT 
Substituting these values in equation (i) we have 

2xy : =KB—c 2 
and 



c 2 



xy=~ 



ENVELOPES. 



223 



which is the equation of a hyperbola referred to its center 
and asymptotes. The curve EF (Fig. 49) is then a hyper- 
bola, and the axis of the ellipse are its asymptotes. 

(124) Let AB (Fig. 48) and AC be the coordinate axes, 
and let the line de of a given length move in such a manner 
that its extremities shall be at all times in the axis. What 
is the equation of the envelope described by that line ? 

Call the length of the line c, and the distance Kd and 
A/ respectively b and a. Let Am=x and mn^y, then the 
general equation of the line will be 







x y 
cT~ b 


= 1 






(1) 


we have also 




a 2 +b 2 


=c 2 






(2) 


Differentiating 


these 


equations 


with 


respect 


to 


a and b as 


variables we have 
















db a 
da b 


b 2 x 
a 2 y 








or 















b 



b 2 X 



a° 



Substituting this value in equation (1) we have 

b 2 x 

■1 



x 
a 



a° 



whence we obtain 



and similarly 



6 / 

a— v 



c'x 



--VcH 



whence 



from which 



i*+b 2 ={c 2 xy+(c 2 yy=c 2 ={c*) 



2l 2. 

x 3 +y 3 '- 




which is the equation of the envelope. 



224 



DIFFERENTIAL CALCULUS. 



The first differential coefficient of this equation is 

_jl X 
dy x 3 jy 3 

dx 



y ° x° 

from which we learn that the curve is tangent to both 
coordinates. 

(125) Suppose the line DC (Fig. 52) to revolve about the 
point D in the axis of abscissas, 
varying in length so that the 
extremity C shall be at all times 
in the axis of ordinates, required 
the envelope described by the 
line DE perpendicular to DC at 
the point C in the axis of ordi- 
nates. 

Representing the distance AD by c, and the tangent of 

1 . 

the angle CDB by — — , its equation will be 




y- 



'-——(x—c) 



in which if we make x=o we have 

c 



y- 



a 



lor the distance from the origin at which the line DC inter- 
sects the axis of ordinates. And since the perpendicular 
passes through the same point, its equation will be 



c 

-ax-\-~ 
a 



(0 

If we consider a in this equation as an independent variable, 
it will represent all the perpendiculars that can be drawn 
under the given condition. 

Differentiating it with respect to a we have 



x— 



ENVELOPES. 



225 



whence 

a= V- x 

and substituting this value of a in equation (1) we have 

whence 

which shows the envelope to be a parabola of which D is the 
focus. It also demonstrates a well known property of the 
parabola, namely, if lines be drawn from the focus perpen- 
dicular to the tangent they will intersect it on the perpen- 
dicular to the axis through the vertex. 

(126) Let AB and EO (Fig. 53) be the coordinate axes, 
and let CD be a line revolving 
between the lines AH and BK 
in such a manner that its ex- 
tremities C and D shall always 
be in those lines, and the pro- 
duct of the distances CA and ~" 
DB from the axis shall be a 
constant quantity. Required the equation of the envelope 
generated. 

Let OA=OB=//z, and AC . BD — c % . Then producing the 
line DC until it meets the axis of abscissas at S, and mak- 
ing the tangent of BSD=<?, we have 

SB : BD : : SO : OF 




or 



whence 



and similarly 



OF _ OF _ 

— +m:BD:: — : OF 

a a 



BD=OF+^ 



AC=OF-am 



226 DIFFERENTIAL CALCULUS. 

whence 

BD . AC=c 2 =OF 2 —a 2 m 2 
or 

But the equation of the line CD is 

y=-ax~\-b 
in which b is the distance from the origin to the point where 
the line cuts the axis of ordinates, that is, the distance OF. m 
Hence 

y=ax + (a 2 ?n 2 +c 2 2 (i) 

is the equation which, when a is variable, represents the line 
CD in every position it can assume under the given condi- 
tions. 

Differentiating with respect to a, we have 

m 2 ada 

xda-\- T~° 

(a 2 m 2 + c 2 ) 2 
whence 

x(a 2 m 2 +c 2 ) 2 

a a * 

mr 

c x 



( m 2 — x 2 ) 2 



which being substituted in equation (i) gives 

c x 2 ( c 2 x 2 a i 

y = — ' — t+(-i l+^Y 

\mr — x A y 



whence 

whence 
whence 



i c 

y[m 2 — x 2 ) 2 = — — x 2 -\-cm 



my{m 2 — x 2 ) 2 =c(?n 2 — x 2 ) 



m 2y2 z= m 2 C 2 — C 2 X 2 



ENVELOPES. 227 

or 

vry 2 -\-c 2 x 2 =m 2 c 2 

which is the equation of an ellipse referred to i s center as 

the origin, and whose semi-axes are m and c. 

(127) The equation of the normal line to the parabola is 

f 

y-y=-j(.X-X f ) (l) 

in which x and y are the coordinates of the point in the 
curve from which the normal is drawn, and x and y are the 
variable coordinates of the normal itself. 

If we consider x and y as variables, equation (1) will 
represent the entire system of normals which can be drawn 
to the parabola. To find this envelope of this system we 
find the relation between x and y from the equation of the 
parabola 

y 2 =-2j)x (2) 

and substitute in equation (1) the value of x\ which gives 

y X y 3 

y-y =-y+^ 

whence 

2j> 2 (y—y f ) = — 2pxy -\-y' 3 (3) 

Differentiating this equation with respect toy only we have 

— 2p 2 =- — 2j>x + 3 j/ % 
whence 

y=v$(px-j>*) 

Substituting this value in equation (3) we have 
whence 

whence 

p*y=-\_\{px-p*yf=-{\pf{x-pf 
whence 

p i y s =^p^{x-py 



2 28 DIFFERENTIAL CALCULUS. 

or 

which as we have seen (Art. 116) is the equation of the 
evolute of the parabola. 

Hence all normal lines to the parabola are tangent to the 
evolute 



SECTION XV. 



APPLICATION OF THE DIFFERENTIAL CALCULUS TO 
THE DISCUSSION OF CURVES. 

THE CYCLOID. 



( 1 28) The cycloid is a curve described by any point in 
the circumference of a circle as it rolls along a straight line 

If for example, the circle EFD (Fig. 54) should roll along 
the straight line AB, 
the point F, starting 
from the point A, 
would describe the 
cycloid AD'B, and 
the distance from A 
to B where the gen- 




G 



E E' B 

Fig. 54- 

erating point again meets, the line AB will be exactly equal 
to the circumference of the generating circle. 
If we place the origin at A we shall have 
KQx—x and FG=y 
The arc FE will be equal to the line AE, and HE will be 
the versed sine of the same arc. Making DE = 2r we shall 
have 

FH 2 =DH . UE=y{ 2 r-y) 
229 



230 DIFFERENTIAL CALCULUS. 

hence 



whence 



FH = GE = arc FE— x = ^ 2 ?y—y 2 

-1 

x=vqi\ sin. y— \/ 27 y— ^2 ^\ 

which is the equation of the cycloid. 

The line AB is called the base of the cycloid, and the line 
D'E' perpendicular to the base at its middle point is the 
axis, and is equal to 2r. 

Since every negative value for y gives an imaginary value 

for x, the curve has no point below the base. If we make 

j = 2r we have 

-1 
x : =VQr sin. 2r=nr 

and every value for y greater than 2r gives an imaginary 

value for xj hence the greatest value of y is the diameter 

of the generating circle ; and for all values of y between zr 

and zero there will be a real value for x. 

( 1 29) We will now proceed, with the aid of the differen- 
tial calculus, to investigate the properties of this curve in 
reference to its tangent, subtangent, normal, subnormal, 
curvature, involute, etc. 

Differentiating equation (1) we have 

rdy rdy—ydy ydy 

dx=-j= — - — = •=—— — 

V 2ry— y 2 V 2ry— y 2 y 2ry— y 2 (2) 

Substituting this value of dx in the general formula for the 
subtangent (Art. 52) we have 

V 2 



V 2ry — y 2 
and for the tangent (Art. 53) 



j 4 



V -^ 2iy — y 2 
For the subnormal (Art. 54) 

GE — y 2ry — y 2 



DISCUSSION OF CURVES. 211 



and for the normal (Art. 55) 



FE= y\/i+ 2/y / =V 2?y 

Since GE the subnormal is equal to ^2/7 —y 2 , which is 
equal to ^/DH.HE, the point E of the subnormal for the 
point F of the curve, must be at the intersection of the ver- 
tical diameter of the corresponding generating circle with 
the base ; and the normal line =<\/l>ry = VDE. Efl must be 
a chord of that circle joining these two points. 

The tangent being perpendicular to the normal will of 
course be the supplementary chord of the same circle. 
Hence to obtain the normal and tangent lines for any given 
point of the cycloid, construct the generating circle for the 
diameter D'E' erected at the middle of the base, and 
through the given point draw the line FH' parallel to the 
base intersecting the circle at F'. Join this point with the 
extremities of the diameter D'E', and the line F r E' will be 
parallel to the normal, and F'D' will be parallel to the tan- 
gent. Hence lines parallel to these, through the given point 
will be the lines required. 

If it is required to draw a tangent parallel to a given line, 
first draw a chord from D' parallel to the given line, and 
through the point where it meets the circumference of the 
circle draw a line parallel to the base. The intersection of 
this line with the curve of the cycloid will be the point of 
tangency. 

(130) From equation (2) we have 

dy y 2ry— y 2 __ / 2r 

dx~ y ~v y ^ 

which becomes zero when j = 2r, hence the tangent at the 
extremity of the axis is parallel to the base. If we make 
y=.o we have 

dy _ 
dx ~~ 



232 DIFFERENTIAL CALCULUS. 

hence the tangent at the base is perpendicular to it. 

Differentiating equation (3) we have 

zrdy rdy 

d 2 y __ y 2 _ y 2 rdx 

dx " 2 / ^ " " dy_ ~ " y 2 

\Z~i~ 1 dx 

hence 

d 2 y _ r 

dx 2 y 2 

This second differential coefficient being essentially neg- 
ative, shows that the curve is everywhere concave toward 
the base. 

(131) The formula for the radius of curvature (Art. 105) 
gives in this case, 

2rdx 2 n A (2rdx 2 \ 2 
(dx 2 + —dx 2 ) 2 o H H 

v y ' \ y J 3.1JL 

T? — — ■ — o * y * v * 

rdx^ rdx 2, ^ 



or 



y2 y% 



R — 2\/ 2ry 



But we have found (Art. 129) the normal to be equal to 
<\/~2ryj hence the radius of curvature at any point is equal 
to twice the normal at that point. Thus at A the radius of 
curvature is nothing, while at D' it is equal to 2D'E'— /\r. 

( 1 32) The equation of the evolute will be found by the 
rule given in Art. 114. 

In the equations of condition (Art. 105) 

00-a—— fc(j—t) (2) 

and 

dx 2 -\-dy 2 

y~ b =- jT y (3) 

Substitute the values of —j- and -^ just found from 



DISCUSSION OF CURVES. 



the equation of its cycloid, and then, by means of that 
equation eliminate x and y. 
Thus 



and 



x—a- 



V 2ry — y 2 



y 



(y-b) 



y—b= 



dx 2 +dx 2 ( — — i j 



dx 2 r 



whence 



or 



y— b=2y and x— #= — 2\/ 2 



ry — y A 



y — —b and x : =^a—2^ — 2 rb—b 2. 
Substituting these values of x andjy in equation (i) (Art. 
128), we have 

a— 2 V _ 2r b —ft =ver. sin." 1 —b— ^/_ 2r b—b 2 
or 

a=VQi. sin.- 1 — £+V — 2 rb— b 2 (4) 

which is the equation of the evolute. 

(133) For all values of b that are positive a is imaginary, 
hence no part of the curve is above the base of the invo- 
lute. For all negative values of b greater than 2r, a is also 
imaginary, hence if we draw A'B' (Fig. 55) parallel to the 
base at a distance 
below it equal to 
2r, the evolute 
will lie between 
that line and the k 
base. If we make 
b = — 2r, a be- 
comes equal to , N 
the arc whose 
versed sine is —b, 




234 DIFFERENTIAL CALCULUS. 

that is half the circumference of the generating circle. 
Hence the point G where the e volute meets the line A'B' is 
in the prolonged axis of the involute. If we make 
b~ o, a also becomes equal to zero, and hence the e volute 
passes through the origin at A, and also the extremity of the 
base at B. For ver. sin."" 1 *? may be zero, or it may be a 
whole circumference. 

If we differentiate equation (4) we have 

rdb rdb+bdb (2r+b)db 

da — — , - — , =a — — , ~ 

V — 2rb— b 2 V— 2rb—b 2 V —2rb—b 2 

or 



db__ V- 2 rb-b 2 

da 2r-\-b 

showing that at the points C and B where <£=<? the base of 
the involute is tangent to the evolute. Also since 

V-2rb-b 2 b 



2r+b v / — 2 rb^b 2 

if we make b= — 2r we have 

db 
— = 00 

da 

showing the tangent to the evolute at G is perpendicular to 

the line A'B'. 

db 
Squaring the value of -j-, and differentiating, we have 

db* b d 2 b r 

and 



da 2, 2r+ b da 2 (2r-\-b) 2 

which is essentially negative, and since every real value of 
b is also negative, the curve is everywhere convex to the 
base of the cycloid. 

( 1 34) These circumstances, together with the form of the 
equation of. the evolute, lead us to suppose it to be an equal 
cycloid, but for certainty we will transfer the origin to G, 
and the coordinate axes to EG and GF respectively par- 



DISCUSSION OF CURVES. 235 

allel to the first. Calling the new coordinates x and y' we 

have 

a=x'-}-m and b—y' + zi 

m and 11 being the coordinates of the new origin referred to 
the original axes. Then 

m=vtr. sin."" 1 2?* and n^ — zr 
whence 

<2~3/+ver. sin. _1 2r and &=— -(2?*— y r ) 

Substituting these values of a and b in equation (4) we have 
x'+ver. sin. _1 2r = ver. sin. _1 (2r— y' ) + \/ 2r(2r— /)— (2r— y')* 
but 

ver. sin."" 1 2/-— ver. sin. _1 (2r— y)=ver. sin. _i y 
hence 

x = — ver. sin . _ x y + V 2 ry —y^ 
which is the equation of the curve CG, the values of x be- 
ing the same as those of x in equation (i) (Art. 128), except 
that they are negative as they should be, since the values 
of x are reckoned in a contrary direction from those of x ; 
and the curve CG is equal to the curve CF, but reversed in 
position with reference to the origin. 

Since the curve CG is equal to FG (Art. 113) the length 
of the cycloid is equal to four times the diameter of the 
generating circle. 

( I 35) The character of the evolute of the cycloid may be 
demonstrated geometrically thus : 

Let us suppose two right lines AB and A'B' (Fig. 55) to 
be drawn parallel to each other, and two circles to be des- 
cribed on the diameters DE and CE, each equal to the 
distance between the two parallel lines and tangent to each 
other at the point C. If now we suppose each circle to roll 
along the line on which it stands, at the same rate, so that 
they are at all times tangent to each other, then the point 
C of the upper circle will describe the first half of a cycloid 
CPF, while the same point C of the lower circle will des- 
cribe the last half of an equal cycloid CP'G. 



236 DIFFERENTIAL CALCULUS. 

Suppose the two circles to have arrived at the point C in 
the line AB, and that P is a point in the upper curve. The 
diameter DC of the upper circle will have assumed the 
position PO, and the diameter CE of the lower circle will 
have assumed the position O'P' parallel to it; and P' will 
be the generating point of the lower cycloid. 

Draw the chord PC' and it will be normal to the upper 
cycloid (Art. 129). Draw also the chord C'P', and it will 
be tangent to the lower cycloid at the point P' (Art. 129). 
Now since PO and O'P' are parallel, these two chords and 
the corresponding arcs are equal, and hence the angles 
PCD' and P'C'E' are equal; and since D'E' is a straight 
line P'C'P is a straight line also, normal to the upper curve 
and tangent to the lower one. Hence the lower cycloid is 
the evolute of the upper one. 

( 1 36) The equation of the evolute may also be obtained 
by considering it as the envelope of the normals drawn to 
the curve. 

The general equation of the normal to the cycloid is 

V 2?y — y * 
in which x and y are the coordinates of that point of the 
cycloid to which the normal is drawn ; and x and y the gen- 
eral coordinates of the normal line. If we make x andj/ 
variables, still retaining their relative values, as in the equa- 
tion of the cycloid, the equation (1) will represent the whole 
system of normals that can be drawn to the curve. If now we 
eliminate one and make the differentials of the equation 
with respect to the other equal to zero, then (Art. 120) by 
eliminating that we shall have an equation which will be 
that of the envelope of the normals, and also the evolute of 
the cycloid. 

Substituting for x in equation (1) its value taken from 
the equation of the curve (Art. 128) we have 



DISCUSSION OF CURVES. 237 



y 

y — r y= 7 f i^ (^"~ ver - sin."" i y+ V 2ry — y 2 ) 

V 2ry — y 2 



or 



— y x+yvQX. si . y , 
2ry — y 



y-y = — w • / rv ~ "^ 



whence 



y\ 2ry — j^' 2 — y ver. sin. _i y +^y =0 (2) 

Differentiating this equation with respect to y' we have 






V 2^y — j> 4 V 2ry — y rf 

Substituting in this equation for ver. sin. _i y its value taken 
from the equation of the cycloid, and multiolying by 
V 2ry — y' 2 , we have 

y(r—y')-(x + V 2 ry '-y 2 )V 2ry -y 2 —yr+x*/ 2 ry -y' 2 ~° 
or 

y( r —. y ) =^V 2ry — y 2 + 2r y —y 2 +yr—x<\/ 2r y _y 2 

or 

X^--y)=---(^--^ / )V2ry--y 3 +3o ;/ --y 3 

but 

1/ — 1/ . . 

x—x'= — ; — V 2ry — y 2 

hence 

y — y 

y(r-yl=~y-(2ry-y' 2 )+ 3 r/--y 2 

clearing of fractions and multiplying we have 

ryy — yy 2 ' == -2ryy — 2ry 2 — yy s -\-y 3 + *ry r 2 — y r 3 
whence 

ryy'— — ry 2 or y — — y 
Substituting this value of y in equation (2) we have 



or 



yV — 2ry— y 2 -\-y ver. sin. -1 — y — xy=o 
#=ver. sin.~ 1 —y+V — 2ry—y 2 



2 3 3 



DIFFERENTIAL CALCULUS. 



which is the equation of the envelope of the normals, and 
also of the e volute of the cycloid, as in Art. 132 ; for sub- 
stituting the variables a and b for x and 7, the equations are 
identical. 



THE LOGARITHMIC CURVE. 



( 1 37) The logarithmic 
curve is one in which one 
of the coordinates is the 
logarithm of the other. 

Its equation is 

^^Log. y 
If we represent the base of 
the system by a the equa- q 
tion may be written 
y=a x 




£-♦-3-2.-/ API 

Fig. 56. 



r ^ 5 b 



The curve may be constructed by laying off on AB (Fig. 
56) the axis of logarithms, the numbers 1, 2, 3, 4, etc., on 
both sides of the origin, and laying off on the corresponding 
ordinates, or on AC the axis of numbers, the corresponding 
powers of a. 

When x=o then y = i, whatever may be the value of a, 
and hence all logarithmic curves will intersect the axis of 
numbers at a distance from the origin equal to 1. 

If a is greater than 1, and x positive, y will increase as x 
increases, and there will be a real value of y for every value 
of x as in the curve DE. 

If x is negative, then the value of y is fractional, and 
decreases as x increases negatively, but y will not become 
zero until #=—00. 

If y is negative, there is no corresponding value of x, and 
hence the curve can neve*r pass below the axis AB. 

If a is less than 1, then y will diminish as x increases 



DISCUSSION OF CURVES. 239 

positively, and becomes zero when #= oo ; but y increases 
for negative values of x, and the curve has a position the 
reverse of the first as DD" in the figure. 
(138) If we differentiate the equation 

y=a x 
we have 

dx m ^ m 

and 

d 2 y 1 1 

v 9 =a x — 2 = y — 2 
dx~ m* J m i 

dy 

If we make y=o we have -r~=o, hence the tangent for 

that value ofy is the axis of abscissas ; and since 7=0 gives 

x=— 00 the axis of abscissas is an asymptote to the curve 

dy 
(Art. ^). But since y= oo gives x= oo, and also ~j~= oo, 

the curve has no tangent parallel to the axis of ordinates 
except at an infinite distance. The sign of the second dif- 
ferential coefficient shows that the curve is at all times con- 
vex toward the axis of abscissas. 

The subtangent PT=-t-j=M; hence the subtangent is 

constant and equal to the modulus of the system of logar- 
ithms, to which the curve belongs. 

In the Naperian system the modulus is i, and in this case 
PT and DA are equal. 

( I 39) We will now investigate the curve whose equation is 

y = x log. x 
Every value for x gives a single value for y. 
If x is less than 1 the value of y is negative. (i) 

If x is greater than 1, y is positive. (2) 

If #=0, or x=i, y z =o. (3) 

If x is negative, y is imaginary. (4) 



240 



DIFFERENTIAL CALCULUS. 



If we differentiate the equation we have 

J*. 

dx 

d 2 y __ 1 
~ x 



log. ^ + 1 



dx 2 



dy 



Making "77 = o we have 



log. #=— 1 or #=£" 



-i-i- 



(5) 

(6) 
(7) 



<? 2.7l82 

which corresponds to a minimum as shown by the positive 

a * y 



value of 



dx 2 " 



When ^=0, 






(8) 

(9) 

(10) 



When x = i, ~r~ — 1 • 

Since y is negative between ^=0 and x = i and then pos- 



itive, while 



^ 3 



is always positive, the curve is concave 



toward the axis of abscissas between x=o and #=1, and 
afterwards convex. ( XI ) 

Hence the curve begins at the 
origin (Fig. 57) and intersects the 
axis of abscissas at D, making 
AD = i (3). The tangent to the 
curve at D makes an angle of 45 ° 
with the axis of abscissas (10), 
while at A the axis of ordinates is 
tangent (9). At the point E, whose 

abscissa is — — ^— , the tangent to the curve is parallel to 

the axis of abscissas (7), and the OTdinate is at a minimum 
(or negative maximum) (8). Between A and D the curve is 
below the axis of abscissas (1), and concave to it (11); and 




Fig- 57- 



DISCUSSION OF CURVES. 



241 



beyond D the curve lies entirely above the axis of abscissas 

and is convex to it. Since x= ^7— gives v 1 ^— x. we have 

2.7182 b -* 

FE=AF 
( 1 40) We will next take the equation 



Every value of x gives a real positive value for jy, and 
hence there can be no negative value of y. (1) 

If x=o, y—o, and hence the curve passes through the 
origin. (2) 

If x is negative we have y z= ^ x i in which if x=o, y will 
become infinite. (3) 

So that x=o gives two values for y, according as x ap- 
proaches zero from the positive or negative side. (4) 

If x be negative and increase in value, that of y will 
approach more nearly to 1, which it will reach when 
*=-oo. (5) 

If x be positive, and increasing the value of y approaches 
more nearly to 1, which it reaches when #=00. (6) 

Hence the curve will be as in 
Fig. 58, in which AB and AC are 
the axes of coordinates, and DE a 
line parallel to AB at a distance 
from it equal to 1. It will pass 
through the origin A (2), extend 
indefinitely in a positive and nega- 
tive direction, and the line DE will 
be an asymptote to both branches (5), (6). The axis of 
ordinates will also be an asymptote in the positive direction 
(3) (Art. 88). As the branch DC of the curve extends to an 
infinite distance in both directions, it has no connection with 




242 DIFFERENTIAL CALCULUS. 

the branch AE, which commences at the origin and is infi- 
nite at the other extremity. There are, in fact, two curves, 
one answering to the positive, and the other to the negative 
value of x. 

If we differentiate the equation y=e x we have 





dy e x 
dx~~ x 2 


and 






jl 




d 2 y e - x (\ — 2x) 




dx 2 ~ x^ 


Since 


l 

e x i 




x 2 ~~ 1 
x 2 e x 


we shall have 






dy 

dx 



when either x=o or #=oo, hence the axis of abscissas is 
tangent at the origin, and parallel to the tangent at an infi- 
nite distance in either direction; in which casej^i (5) (6). 

d 2 y . 
For all negative values of x, , « is positive, and hence 

the branch DC is convex to the axis of abscissas. For all 

d 2 y . 
positive values of x less than -J-, , ^ is also a positive, show- 
ing that between A and H the curve is convex to the axis 
of abscissas, while at the point H, where x=^, the value of 

d 2 y . . . .11 

~7~y changes from positive to negative passing through zero, 

showing that at P the curve ceases to be convex, and be- 
comes concave toward the axis of abscissas. This is called 
an inflexion. 



SECTION XVI. 



SINGULAR POINTS. 

(141) Singular points of a curve are those at which there 
exists some remarkable property not common to other points 
of it. Such, for example, as the maximum or minimum 
value of the ordinates or abscissas, points of inflexion, con- 
jugate points, cusps, etc. 

In many cases these points are easily discovered by the 
aid of the differential calculus, as will be seen by the fol- 
lowing examples. 

MAXIMA AND MINIMA. 

(142) If we differentiate the equation 

j/=3 + 2 (*-4) 4 = 
we shall have 

^=48(^-4) 

Here we find that x=4 will reduce the first differential 

243 



244 



DIFFERENTIAL CALCULUS. 



coefficient to zero, showing that the tangent to the curve is 
parallel to the axis of abscissas (Art. 36), and hence the 
value of the ordinate may be a maximum or minimum. But 
since the sepond differential coefficient is always positive 
except when it is zero, the first must be an 
increasing function, and hence at zero must 
be passing from negative to positive, and the 
value of y must be changing from a diminish- ■ 
ing to an increasing one. So that there is a 
minimum when #=4, as shown in Fig. 59. lg ' 59 * 

We infer the same- thing from the sign of the fourth differen- 
tial coefficient (Art. 29). 
If we take the equation 

y = 2 — 2{x — 2) 4 




we shall have 



f=-8(*- 2 )a 

d 2 y 

-^=- 24 (x- 2 y 
d 3 y 

d^y _ 
dx* ' 



=-48 



Since .#=2 reduces the first differential coefficient to zero 
the tangent at that point is parallel to the axis 
of abscissas; and since the fourth differential 
coefficient (the first that has real value for 
#=2) is negative, the value of y at that point • 
must be a maximum as in the figure. Since 



c 

Fig. 60. 



•y . 



■-r^ - is at all times negative, except when #=2, the curve 

will be concave tow r ard the axis of abscissas for all positive 
values of y (Art. 91). 



SINGULAR POINTS. 245 



POINTS OF INFLEXION. 



(143) A point of inflexion is one in which the radius of 
curvature changes from one side to the other of the curve 
so that it will be convex on one side of the point of inflex- 
ion, and concave on the other towards any line not passing 
through the point itself, and this will, of course, be true for 
the axis of abscissas, and hence at such a point the second 
differential coefficient will change its sign. If the point of 
inflexion should be in the axis of abscissas, both parts of 
the curve would be convex or concave to it, but the second 
differential coefficient will still change its sign (Art. 93). 
Now in order that the sign should be changed, the function 
must pass through zero or infinity, and hence the equations 

d 2 y d 2 y 

, 2 — o and , 9 — °° 
dx* dx* 

will give all the points of inflexion in any curve in which 

there may be such points. 

( 1 44) Let us now take the equation 

y=i+3( x — 2 ) 3 
whence 

and 

d 2 y 

^r=i8(*-2) 

d s y __ 
dx 3 
In this case every value of y gives one for x, and vice 
versa, hence the curve has no limit. When #=2, then 

-y-=o and y= 1, so that if we make AC =2 

and CD = i (Fig. 61), the tangent at D will 

be parallel to the axis AB. But x=2 re- 

d 2 y 
duces -r-gT to zero, also indicating that Fig. 61. 



LZ. 



246 DIFFERENTIAL CALCULUS. 

there may be a point of inflexion, hence we resort to the 

d 3 y 

value of -7-3- which is a positive constant. From this we 

d 2 y ... 
learn that at zero , 2 ls an increasing function, hence it 

must pass from negative to positive, showing that at the 

dy ... 

same point ~r~ passes from a decreasing to an increasing 

function, and hence does not change its sign, but remains 
positive both before and after the zero point; and this shows 
that the value of y is an increasing function both before and 
after the same point. There is, therefore, no maximum nor 
minimum for it at that point. 

d 2 y 

Since -j-jf changes its sign at x=2 from negative to pos- 
itive, the curve will be concave toward the axis of abscissas 
when #<2, and convex when x>2 y so that at the point D 
where x = 2 the curvature changes its direction and there is 
an inflexion. 

(145) If we take the same equation, and make the last 
term negative, we shall have 



d 2 y 
■£=-i8(*- a ) 



D 



A C B 

dx A V Fig. 62. 

The point D where x — 2 will still be the point of in- 

dy . . 

flexion, but since -7- is negative for all values of x 

except #=2, the curve will approach the axis of abscissas 

d 2 y 
for all positive values of y, except y = i 9 and since ~^j- 

is positive for x<2, and negative for#>2, the curve will be 
convex toward AB between A and C, and concave afterwards, 
as in Fig. 62. 



SINGULAR POINTS. 247 

The first differential coefficient being zero when #=2, it 
follows that the tangent at that point will be parallel to the 
axis of abscissas (Art. 36), and hence the curve will pass 
from one side of the tangent to the other at the point of tan- 
gency, and will be convex to the tangent on both sides of it. 

(146) If we take the equation 

y = 2-\-2{x — 2) 5 
we have 

dy__ 6 

dx ~ s { x - 2 y 

d 2 y 12 



dx 2 f JL 

25(^-2)* 

If we make x= z 2 we have 

dy d 2 y _ 



y= 2 > ^r=°° 



2 




' dx ' dx 

and hence at the point D where x = 2 the tangent will be 

perpendicular to the axis of abscissas (Fig. 63), 

dy . 
and since ~r~ is positive for other values of x y 

the curve will leave the axis of abscissas, for all 

positive values of y as x increases. And since 

d 2 v . . . . . 

-j-ijr changes its sign from positive to negative m passing 

through infinity where x=2, the curve will be convex toward 
the axis of abscissas for x<2, and concave for x>2, and at 
x = 2 there will be an inflexion. 

(147) If in the same equation we make the last term 
negative we have _3 

y=-2 — 2(x— 2) 5 
and 

jfy_ 6 

dx ~ ( >,-! 

5(^— 2 ) 
d 2 y 12 



2 s( x — 2 ) T 



248 



DIFFERENTIAL CALCULUS. 



C B 

Fig. 64. 



and the conditions will be changed so that the curve will be 
reversed. It will now approach the axis of 
abscissas, and the second differential coeffi- 
cient will change its sign from negative to 
positive in passing through infinity where 
x=2; the curve will be concave for x<2, 
and convex for x>2. 

The point D (Fig. 64) will still be a point of inflexion, 
and the tangent will be perpendicular to AB. 

( 1 48) If we take the equation 
y=(x— 2) 3 
we have 

and 

^=6(*- 2 ) 

which all reduce to zero when x — 2. 

This shows that the curve meets the axis of abscissas 
the point where x=2, and that this axis is 
tangent to it there. And since the second 
differential coefficient will have the same 
sign as y (both being the same as that of 
x—2), it will change from negative to posi- 
tive at the point where ^=2, showing an inflexion there, and 
that the curve is convex to the axis of abscissas on both 
sides of it. 



at 



A 



7^c 



Fig. 65. 



CUSPS. 



( 1 48) A cusp is a curve consisting of two branches start- 
ing from a common point in the same direction, and imme- 
diately diverging from a common tangent. They are of two 
kinds, namely : Those in which the branches are on differ- 



SINGULAR POINTS. 249 

ent sides of the tangent, which are cusps of the first order ; 
and those in which the branches are both on the same side 
of the tangent, which are cusps of the second order. 

The following are examples of the first order. 

Let 

then 

d y ( \-\ 

and 

d 2 y _ 2 



dx 2 



If we make x=i we have 






dy d 2 y 

'=*> ^=°°' and ^"=-° c 

dy , 
For every value of #<i, -j- is negative, and positive for 

every value greater. The curve, therefore, approaches the 

axis of abscissas, in the first case, and recedes from it in the 

second {y being positive), which indicates a minimum, while 

the tangent at that point is perpendicular to 

d 2 v . 
the axis of abscissas ; and since , ^ is always 

negative the curve is concave toward the same A C B 
axis. Fi §»- 66 - 

Every value of y less than i gives an imaginary value for 
x, while every value greater than i gives two values for x, 
one less and one as much greater than i. Hence the curve 
has two equal branches commencing at D (Fig. 66), where 
they have a common tangent. 

(150) If we make the last term negative, the signs of 

dy d 2 y 

-j- and -r-f will be reversed, and the first will change from 



4- 



250 DIFFERENTIAL CALCULUS. 

positive to negative as x passes from.r<i to 
x> 1 ; while at x=i the tangent is still perpen- 
dicular to the axis of abscissas. Any value of y 
greater than 1 will give an imaginary value for At C R 
x, while every value less than 1 will give two Fl &- 6 7- 
real values for x equally distant from the point C where 

d 2 y 
#=1 (Fig. 67). The sign of ~T~% being now always posi- 
tive (except at x=i) shows that both branches of the curve 
are convex toward the axis of abscissas. These are then 
cusps of the first order. 

(151) If we differentiate the equation 

3 

y = 2 ±(x— i)"* (i) 

we have 

ax * v J 

We see from equation (1) that when x=i,y=2, and when 

#< 1, y is imaginary, while when x>i,y has two values, one 

greater than 2 and the other as much less; so that DC (Fig. 

6S) be drawn perpendicular to AB, making DC=2, the curve 

will commence at D and be symmetrical about the line DE, 

dy ... 

and since -j- =0 for the point D, the line DE will be tangent 

to both branches. Since for every other value of 

dy . E 

x, —r~ has one negative and one equal positive 



value, one branch of the curve will approach the A C B~ 

axis of abscissas, and the other recede from it at Fi §- 68 - 

d 2 y 
an equal rate. And since for every value of #>i, , 2 

has two equal values with contrary signs, the positive cor- 
responding with the greatest value of y, we infer that the 



SINGULAR POINTS. 25 I 

upper branch of the curve is convex, and the lower branch 
concave, to the axis of abscissas, and that the curve is a 
cusp of the first order. 

(152) If we change the sign of the last term and make 
the equation 3 

J/ = 2±(l— x) 2 

we have 



dy 
dx 
d 2 y_ 



= ±1(1-.)* 



:±- 



** 4(1 -*)* 

and the curve will be similar, but reversed in position as in 
Fig. 69. 

If #>i, y will be imaginary. 

If x=o, y = 3 and y=i. 



>f 



C 



If y=o, #=i — y^. Since , 2 is both pos- A 

itive and negative when x<i there is no maxi- Fig. 69. 
imum nor minimum value for y. 

(153) The curve represented by the equation 
(y— x 2 ) 2 = x 6 
contains a cusp of the second order, as well as some other 
singular properties. 

Solving this equation we have 

y—x 2 dtx 2 (1) 



and by differentiation we have 

dx 2Xj -2 






and 



J 2 y is * 

— — 2 1 X 

dx 2 ~ 4 



From equation (1) we find that the curve passes through 




252 DIFFERENTIAL CALCULUS. 

the origin, and does not extend to the 

negative side of the axis of ordinates. 

Every positive value for x<i gives two 

real positive values for y, while x=i 

gives one positive value for y and one Al 

equal to zero. Hence the curve has 

two branches, both of which pass 

through the origin, and one intersects the axis of abscissas 

at a distance from the origin equal to 1. 

dy 
If we make -t~= o we have x=o and x=^. Hence 

there are two points in which the tangent to the curve is 

parallel to the axis of abscissas ; at the origin where the 

axis itself is tangent and at the point D (Fig. 70) whose 

abscissa is #=Jf ; and as the value of x at this point 

. dy 
derived from the equation -7- =0 corresponds to the minus 

sign in equation (1), the point of tangency is on the lower 
branch of the curve. 

15 1 

The second differential coefficient has two values 2+ — x 2 

4 

and 2 — — x 2 , of which the first belongs to the upper branch 

of the curve and is always positive, while the second is pos- 

15 i_ . 
itive so long as — x 2 is less than 2 ; that is so long as x is 

d 3 y 
less than ^^, which makes , 2 ~°- After that it becomes 

negative ; showing that the lower branch of the curve is 
convex to the axis of abscissas, as far as the point whose 
abscissa is AE= f 6 ^, and at this point there is an inflexion, 
the curve becoming concave to the axis of abscissas as long- 
as y is positive and convex afterward. Hence at the origin 
there is a cusp of the second species. 



SINGULAR POINTS. 



253 



CONJUGATE POINTS. 

( 1 54) Conjugate points are those single points which are 
isolated from the curve, but will satisfy the equation. 
If we differentiate the equation 



y=± 



V : 



x 2 (x— b) 



we have 



(1) 



-2b 



dy zx- 

dx~ 2^4^-7) 
Zx—Ab 



d 2 y 
dx 2 " 



4a 2 (a 



bf 



If we make x=o in equation (1) we have y=o, but any 
other value of x less than b will make y imaginary. Hence 
while the origin will satisfy the equation, that point is iso- 
lated, having no connection with the curve. We also see 
that x=o will give 

dy b 



dx ~"y 



- ab 



which is imaginary as it should be, since at that point the 

curve can have no tangent. 

If we make #=£, we have 

dy _ 

dx 

showing that the tangent at that point is perpendicular to 

the axis of abscissas, while the value of 

y is zero. As every positive value of 

x>b gives two equal values for y with 

opposite signs, the curve is symmetrical 

about the axis of abscissas, and as the 

dy 
value of ~r has the same sign as^, the 

uX Fig. 71. 

curve departs from that axis in both directions. If we make 
x negative the value of y becomes imaginary ; showing that 




254 DIFFERENTIAL CALCULUS. 

the curve does not extend to the negative side of the axis of 
ordinates. 



d 2 y 

~^o, we na 



If we make , 2 =o, we have 



x- 
3 

showing that at the points C and C r (Fig. 71) which lie in 
the ordinate drawn through D at a distance from the origin 
equal to f£, the curve has an inflexion in each branch, since 
for that value of x we have 

iT 

^ -*" v 3 tf 

If we make x<- L 7 ? the second differential coefficient will 
o 

4^ 
have a sign contrary to that of y. If x>- — - the signs will 

be the same. Hence between H and D the curve is concave 

toward the axis of abscissas, and convex beyond D, which 

also shows an inflexion. 

dy 
If we make -3—= o we have 3^ — 2^, or 

_2b_ 
X " 3 

This value being substituted for x in equation (1) gives 
an imaginary value for y, showing that there is no point in 
the curve where the tangent is parallel to the axis of abscissas. 

MULTIPLE POINTS. 

(1 55) A multiple point is one in which two or more 
branches of a curve intersect each other. At such a point 
the curve will always have as many tangents as there are 

branches, and hence -7- must have the same number of val- 
ues for that point. 



SINGULAR POINTS. 



2 55 



Let us take the equation 

y=Z>±(x—a)\/x — c where a>c (i) 

tiien by differentiating we have 

dy x— a 

For x=a and x=c in equation (i) we have y—b; hence 
H and H' (Fig. 72) corresponding to 
these values of x and y are points in 
the curve. For all values of x<Cc 
that of y is imaginary ; hence there 
is no part of the curve between H and a 
the axis of ordinates. For every 
value of x>c, except x=a,y has two 
values, one greater, and the other as much less than b. 
Hence the curve is symmetrical about FIH\ For x—c we 

have -7- — 




dx 



= oo, hence the tangent at His perpendicular to 

dy 



the axis of abscissas. For x=a we have two values of 



dx 



equal to each other with contrary signs, namely, V ' x —c and 
— %/x — c. Hence at H' there are two tangents making sup- 
plementary angles with the axis of abscissas, so that the two 
branches of the curve cross each other at that point in direc- 

dy 
tions symmetrical with HH . If we make ~f — o we have 

a+2c 

~7 3 
which shows that at the point corresponding with the ordi- 
nate at E where AE equals one- third of (2AC+AB), the tan- 
gent is parallel to the axis of abscissas. 

(156) We will close the discussion of algebraic curves 
with that of the equation 

ay 2 —x 3 +(b—c)x 2 +bcx=-o 
Solving this equation with reference toy we have 



256' 



DIFFERENTIAL CALCULUS. 



'x(x — b)(x-\-c) 
a 



and 



2,x 2 — 2x(b—c)—bc 



(1) 



± =± 



^Q 



If in equation (1) we make x=o, x=b,or x = —c, we have 
in every case y=o. Hence there are three points, H, A and 
H' (Fig. 73), where the curves meet the axis of abscissas. 

Every negative value of x>c gives an imaginary value 
for jy, hence the curve has no point on the negative side of 
H, since AH =c. Every negative value of x<c will give 
two equal values for y with opposite signs ; hence from H 
to A the curve is symmetrical about the axis of abscissas. 
Every positive value for x<b gives an 
imaginary value for y; hence no part 
of the curve lies between A and H\ 
Every positive value for x>b gives 
two equal values for y with contrary 
signs. Hence on the positive side of 
H' the curve is symmetrical about 
the axis of abscissas, and the entire curve consists of two 
parts having no connection with each other by a common 
point. 

Each of the values of x that reduce y to zero also reduce 

-7- to infinity; hence at the points H, A and H' the tangent 

is perpendicular to the axis of abscissas, and one of these 
tangents is the axis of ordinates. 
If we solve the equation 

$x 2 — 2x{b— c) — bc=o 
we shall have 

_b—c±\/ $bc+(b—cY 




Fig. 73- 



but \/zbc+(b-cY<b+c s 



3 
hence 



if we take the positive 



SINGULAR POINTS. 257 

value of the radical part the result will be less than 

b—c+b+c . 9 ... 

, that is, less than -p, hence it will give no point 

of the curve. If we take the negative value, the result will 
be numerically less than —\c j hence there will be two 
points where the tangent will be parallel to the axis of 
abscissas, corresponding to the point on that 'axis where 



b—c— V $bc + (b—c) 2 

x— 

3 

If £=0 the equation becomes 

ay 2 = x 3 —bx 2 

in which case the oval HA is contracted into a conjugate 
point at A as in Art. 154. 

If b—o the equation becomes 

ay 2 =x 3 -\-cx 2 
or 



v=±\/ 1 



t X 3 +CX 2 

y=±V 

and 

dy _ $x 2 +2cx 
dx~~ 2Vax 2 (x+c) 

In this case the curve takes the form 

in Fig. 74. There are two equal values T 

for y with opposite signs for every value 

of x on the positive side of H where 

. dy 
x=—c. At that point -j- =00, and the 

tangent is perpendicular to the axis of Flg " 74 ' 

dy 2c 

abscissas. If we make ~r-=o we have x~o and x — — — ; 

ax 3 

2€ 

hence the tangent at A and at T and T where x=— — are 

. 3 

parallel to the axis of abscissas. 



^ 



T' 



25$ 



DIFFERENTIAL CALCULUS. 



If we make both b and c equal to zero we have 



whence 



and 



y 



«:V£ 






^ " * * v a 

In this case the curve assumes the 

form in Fig. 75. There is no negative 

value for x, and all positive values of x 

give two equal values for y with contrary 

<iy 

signs. At the origin we have ~r~ ~o, and 

hence the axis of abscissas is tangent to 

both branches of the curve which is a cusp of the first 

species 



Fig- 75- 



PART II. 



Integral Calculus. 



259 



I NTEGRAL C ALCULUS. 



SECTION I. 



PRINCIPLES OF INTEGRA TION. 

(157) The problem of the differential calculus is to 
obtain the differential or rate of change in a function arising 
from that of the variable, or variables, which enter into it. 
The corresponding problem of the integral calculus is to 
pass from a given differential of a function to the function 
itself. 

The first of these operations can always be performed 
directly by rules founded on philosophical principles. The 
second can only be performed by empirical rules founded on 
actual experiment. We cannot proceed directly from the dif- 
ferential to the function, but, as it were, backwards ; that is, 
we show that a function is the integral of a given differen- 
tial by showing that the latter would be produced by differ- 
entiating the former. Thus we know that x 2 is the integral 
of 2xdx, because 2xdx has been shown to be the differential 
of x 2 . Hence the rules for integration are merely the rules 
for differentiation inverted. 

While rules have been obtained for differentiating every 
algebraic function^ it by no means follows that every differ- 
_ _ 261 



262 INTEGRAL CALCULUS. 

ential can be integrated. The number of simple algebraic 
functions is very small, and each one has its specific form 
of differential. Should any function be complicated, it can 
be analyzed and differentiated in detail, applying only the' 
rules for simple forms. But before a differential can be 
integrated, it must be reduced to one of the forms arising 
from differentiating a simple function ; and this can be done 
in comparatively few of the infinite number of forms that 
differentials may assume. The transformations available for 
this purpose form one of the chief subjects that demand the 
attention of the student of the integral calculus. The dif- 
ficulty of integration is very much increased when the dif- 
ferential is a function of two independent variables, for the 
rate of change in such a function can give but little indica- 
tion, generally, w T hat the function is. 

There is still another difficulty in obtaining the exact 
integral of any given differential. We have seen that the 
constant terms in any function disappear when it is differ- 
entiated, and, of course, when we come to integrate an iso- 
lated differential expression, we cannot know what constants, 
if any, should belong to it. In such a case, then, we pay no 
attention to the question of constants. If, however, the 
function should occur in an equation we can generally find 
from the conditions expressed by it what value would belong 
to the constant. Until this is done we indicate by adding 
the symbol C to the integral that a constant is to be supplied 
if needed *to render the integral definite. Until then it is 
said to be indefinite. 

The notation indicating the integral of any differential is 
the letter s elongated, thus Jxdx would be read "the integral 
of xdx" This notation was originally adopted by Leibnitz 
to indicate the sum of the infinitely small differentials or dif- 
ferences of which he supposed the function to be made up, 
and is still retained as a matter of convenience even by 



PRINXIPLES OF INTEGRATION. 263 

those who reject its original meaning, as employed in the 
system of Leibnitz. 

The following^ rules for integration are derived from those 
for differentiating ; being in fact the same rules inverted. 

(158) If the differential have a constant coefficient it may be 
placed without the sign of integration. 

For we have seen (Art. 10) that the differential of a vari- 
able having a constant coefficient is equal to the constant 
multiplied by the differential of the variable ; that is to 
say, the coefficient of the variable will also be the coeffi- 
cient of its differential; hence the coefficient of the differ- 
ential will also be the coefficient of its integral, that is, of 
the variable ; and may be placed outside the sign of integ- 
ration. 

Thus 

d(ax) =adx hence Jadx=qfdx=ax 

(159) The integral of a differential function, consisting of 
any number of terms connected together by the signs plus and 
minus, is equal to the algebraic sum of the integrals of the terms 
taken separately. 

For we found (Art. 9) that the differential of a polynom- 
ial is found by differentiating each term separately, hence to 
return from the differential to the polynomial, which is the 
integral, we must integrate each term separately. Thus 

d[x -\-j — z) = dx +dy— dz 
hence 

f(dx-\-dy— dz) =-fdx -\-fdy —fdz — x +y — z 

(160) The integral of a monomial differential consisting of a 
variable, multiplied by the differential of the variable is equal to 
the variable raised to a power with an exponent increased by one, 
and divided by the increased exponent and the differential of the 
variable. 

We have in (Art. 15) the rule for obtaining the differen- 
tial of the power of a variable. In other words we have 



264 INTEGRAL CALCULUS. 

given the steps by which we pass from the power to its dif- 
ferential ; and hence to pass back from the differential to its 
integral, that is, the power, we must retrace each step. Thus 
in the first case we diminish the exponent by one ; in the 
latter we increase it by one. In the former we multiply by 
the differential of the variable ; in the latter we divide by 
it. In the .former we multiply by the exponent before reduc- 
ing it ; in the latter we divide by the exponent after increas- 
ing it. Thus 

Jux n ~ ^dx — x n 
because 

dx n =ux n ~ 1 dx 
(161) If the function consist of the power of a polynom- 
ial multiplied by its differential, the same rule will apply. 
Thus let the differential be 

(ax+x 2 ) n (a-\-2x)dx=(ax-t-x 2 ) n d(aX'i-x 2 ) 



make 

then 

and 



ax-\-x 2 —u 
(ax-\-x 2 ) n (a-{-2x)dx=u n du 

u n+1 (ax+x 2 ) n+1 

ln n du=- ; — ■ = ; 

** n-f-i n + i 

EXAMPLES. 



x 2 dx 
Ex. 1. What is the integral of ■ ? 



JL 

Ex. 2. What is the integral of x 3 dx? 

dx 
Ex. 3. What is the integral of — 7^=? 

dx 
Ex. 4. What is the integral of ■ — g- ? 



Ans. 


X 6 

9 


Ans. 


4 

$x s 

4 


Ans. 


2\S X 


Ans. 


I 

~~lx~ 2 



PRINCIPLES OF INTEGRATION. 265 

dx 

Ex. z. What is the integral of ax 2 dx+ — 7= ? 

J ° 2y x 

ax 3 __ 

Ans. +\/x 

3 

(I 62) If the exponent of the variable in the case pro- 
vided for in Art. 160 should be — 1, the rule will not apply. 
For by this rule 

r -17 - x ° J 

x l dx — — =— = 00 

J 00 

and this arises from the fact that a differential with such an 
exponent can never occur under the rule given in Art. 15 ; 
for then the variable must have been x°, a constant quan- 
tity, that cannot be differentiated. Such differentials, how- 
ever, do frequently occur, but the rule for their integration 
must be drawn from a different source. We have found 

dx 
(Art. 38) that the differential of log. #= — =x~ 1 dx y and 

hence a differential of this kind must be integrated by the 
rule derived from that given for differentiating logarithms. 
That is to say, the integral of any fraction, in which the 
numerator is the differential of the denominator, is the 
Naperian logarithm of the denominator. 

EXAMPLES. 

adx 
Ex, 1. What is the integral of ? Ans. a log. x 

2bxdx 
Ex. 2. What is the integral of , , 2 ? 

Ans. log. (a+bx 2 ) 

adx a 

Ex. 3. What is the integral of — ; — ? Ans. -7 log. x 

ax 2 dx 
Ex. 4. What is the integral of ^~ ? Ans. a log. x 

dx 
Ex. 5. What is the integral of -77— ? Ans. log. (a+x) 



13 



266 INTEGRAL CALCULUS. 

(163) If the differential be in the form of a polynomial, 
raised to a power denoted by a positive i?itegral exponent and 
multiplied by the differential of the variable, the integral may be 
found by expanding the power and multiplying each ter?n by the 
differential of the variable. We may then integrate the terms 
separately. Thus let us take the expression 

[a-\-bx) 2 dx 
Expanding the binomial and multiplying each term by dx 

we have 

a 2 dx-\-2abxdx-\-b 2 x 2 dx 
which may be integrated as in Art. 158. 

' EXAMPLES. 

Ex. 1. What is the integral of ($+jx 2 ) 2 dx ? 

Ans. 2^x+ 1 ^~x s +^-x 5 
Ex. 2. What is the integral of (a+^x 2 ) 3 dx? 

Ans. a 3 x-\-^a 2 x s -j---£-ax 6 +- T 7 -^ 7 
( I 64) If a binomial differential be of such a form that the 
exponent of the variable without the parenthesis is one less than 
that of the variable within, the integral will be found by increas- 
ing the exponent of the binomial by one and dividing it by the new 
exponent into the exponent of the variable within into its coefficient. 
For suppose the differential to be 

(a-\-bx n ) m x n -\ix 
make 

then 

and 



from which 



a+bx n —p 
dp—nbx n ~ 1 dx 

dp 

x n ~ 1 dx= — 7- 

nb 

p m dp 
(a+bx n Yx^'^dx -—^ff 



PRINCIPLES OF INTEGRATION. 267 

of which the integral is 

pm±l (g+bx n ) m+1 
(m-\-\)nb~ {m-j-i)nb 
hence the rule. 

EXAMPLES. 

1 
Ex. 1. What is the integral of (a+bx 2 ) 2 7/ixdx? 

m 3. 

Ans. ~y(a + bx 2 ) 2 

Ex. 2. What is the integral of (a 2 +x 2 ) 2 xdx? 

Ans. {a 2 +x 2 Y 

Ex. 3. What is the integral of (a+bx 2 ) 2 cxdx? 

c(a + bx 2 ) 2 
Ans. ~ b 

(865) Every rational fraction, which is the differential of 
a function of x y may be put under the form 

Ax m +Bx m ~ 1 +Cx m -*+ D^ + E 

Fx n +Gx n - 1 +Hx"'-2+ Kx+lJ** 

in which the greatest exponent of the variable in the denom- 
inator exceeds by one or more the greatest exponent in the 
numerator. For if it is equal or less, a division may be 
made, until the exponent of the remainder would become 
less than that of the divisor, and this remainder would be- 
come the numerator of the fractional part of the quotient ; 
the other part, consisting of entire terms, would be integrated 
as in Art. 159. Hence we need only to consider the method 
of integrating the fractional part of the quotient, or rather 
any fractions of the form already given. 

(166) For this purpose we resolve the denominator into 
factors of the first degree at 

(x— a)(x— b)(x— -c)(x— d) etc. 
and place the fraction under the form 

/ A B C D 

( + 7+ + -,+ etc.V* 

\x—a x—b x—c x—d J 



268 INTEGRAL CALCULUS. 

in which A, B, C, D, etc., are constants, whose values are 
determined by reducing all the fractions to a common de- 
nominator, and placing the sum of the numerators equal to 
the original numerator (the denominators being identical). 
Since this equality of the numerators must exist, indepen- 
dent of any particular value of x, the coefficients of the like 
powers of x must be respectively equal to each other; and 
this will furnish enough equations to determine the values 
of the constants. Substituting these values the fractions 
may then be integrated separately. 

(167) For example let us take the fraction 

2adx 
x 2 —a 2 ^' 

which by decomposing the denominator may be put into the 
form 2adx 

{x~\-a)(x—a) 
which we transform into 

A B 

^x-\-a x—a) 

which being reduced to a common denominator becomes 

A^— A^+B^+B c? 

(x — a)(x-\-a) 

Making this last numerator equal to that of (i) we have 

2a —Kx — A<z+B.2:+B# 
or 

(A+B)^ + (B-A-2>=o 
from which we obtain 

A+B=o 
and 

B-A-2=o 
whence 

A = — i and B = i 
Substituting these values of A and B in (2) we have 
2ddx dx dx 
x 2 —a 2 x—a x-\-a 



(—77 + )dx (2) 

\x-\-a x — af x J 



PRINCIPLES OF INTEGRATION. 269 

and by integration 

/2adx /» dx r dx 
^n—^J ^T a -J ^T^log. (x-a)-\og. [x+a) 

( I 68) Let us next take the fraction 

a^+bx 2 

n q (ZX 

a^x — x 6 
in which the factors of the denominators are x and (a 2 — x 2 ) 
or x{a-\rx)(a— x). If we make 

a 3 +bx 2 _A _B_ C 
x(a-\-x)(a — x) x a—x a+x ^ ' 

and reduce the second member of the equation to a com- 
mon denominator we have 

a s +bx 2 _ Aa 2 _ Ax 2 +Bax+Bx 2 +Cax-Cx 2 
a s x — x 3 x(a — x){a J rx) 

and placing the coefficients of the like powers of x in the 
numerators equal we have 

B-A-C=£ 
Btf+C#=o 
Aa 2 =a s 
The last of these equations gives 

A— a 
which reduces the first to 

B-C=a+b 
and this combined with the second gives 

a + b a-\-b 

B = and C=- 

2 2 

Substituting these values of A, B and C in equation (i) we 
have 

a^+bx 2 adx a+b a+b 

o CtX ~T~ / \ 11 X / , \ LlX 

a 6 x—x 6 x 2\a—x) 2\a+x) 

and by integration 

/a^+bx 2 a+b a + b 

-^—-^dx=a[og. x---- log. {a-x)—^\og. (a+x) 



270 INTEGRAL CALCULUS. 

which may be reduced to 

a log. x—{a+b) log. (a 2 —x 2 ) 

Note.— The second term of the integral must be negative ; for since d(a—x) is 

dx dx 
—dx, we shall have d{log.{a—x)=——~— and hence —must be the differential 

of —log. (a— x), 

(169) Let us now take the fraction 

xdx 

x 2 -\-<\ax—b 2 
To find the factors of the denominator we must make it 
equal to zero, and solve the equation which gives 

00—— a±V 4 a 2 +b 2 
and hence the factors of the denominator will be 
x+a+'y/^ + frl 

and 

x+a— ^/ 4 a 2 +b 2 
To simplify the expression we will represent the constant 
part of each factor by E and F and we shall have 

x 2 + 4 ax-b 2 ^(x+E)(x+E) 
and we may make 

x __ A B _ A.t+AF+B^+BE 

t 2 + 4 ax-b 2 ~x+E + x + F~ (x+E){x+E) 
making the numerators equal we have 

A^+AF+B^+BE=x 
whence 

A+B = i 
and 

AF+BE=o 
from which 

E F 

A =E3F and B=Z "~E^F 
Substituting these values of A and B we have 

/xdx E P dx F f dx 

x 2 + Aax-b 2 = E-FJ x+E~E-eJ x + F 



PRINCIPLES OF INTEGRATION. 27 1 

which becomes by integrating 

E F 

g— plog. (^+ E )-EZf lo §' (*+ F ) 

or by substituting the values of E and F 



/. 



xdx <^+v / 4 72 ~f b* 

x 2 + 4ax—b 2 ~ 2 V 40 s + frz 



\og\x+a + V 4a 2 +b' 6 ) 



-V 4 a 2 +b 2 , , 

log. (x+a-V 4 a 2 +b 2 ) 



2 V 4a 2 +b 2 

( 1 70) In all these cases the factors of the denominator 
are unequal. If a part or all of them are equal the rule 
will not apply. For suppose we have 

?x±+Qx 3 +Rx 2 +S;c+T 
(x— a)(x— b){x— c)(x— d)(x— e) 
which we make 

A B C D _^ E 

x — a x—b x—cx—d ' x— e 
if some of these factors are equal, say a—b=c, we should 
have 

P^ 4 +etc. _ A+B+C D E 

(x — a)' s (x— d)(x— c) x—a ' x—dx—e 

Thus in reducing the second member to a common de- 
nominator, A+B-f C would have to be considered as a sin- 
gle constant A', and the three constants A', D and E would 
not be sufficient to establish the five equations of condition 
which are required in making equal the coefficients of the 
like powers of x. In order to avoid this difficulty we decom- 
pose the original fraction and make 

~Px±+Qx 3 + etc. _ A+B^+Ct 2 D E 

(x— a) 3 (x— d)(x— c)~ (x—a) s x—dx—e 

which contains the necessary number of constants, and at 
the same time, when reduced to a common denominator, 
will produce a numerator containing x to the fourth power; 
thus giving a sufficient number of equacions between the 
coefficients of the like powers of x. 



272 INTEGRAL CALCULUS. 

In the meantime the expression 

A+Bjc + Cjc 2 
(x— a) 3 
may be put into the form 

A' B' C 

{x—a) 3 (x—a) 2 x—a 
in which A', B' s C' are determinate constants. For let 

x—a—u then x=u+a 
and 

A+Bx + Cx* _A+Ba + Ca 2 +Bu + 2Cau+Cu 2 
{x—a) 3 u 3 

__A+B^fW B + 2O C 
u 3 u 2 u 

and replacing the value of u we have 

A+B*+C*» A+B^+Ca 3 B + 2G* C 



(#— #) 3 (.#— <z) 3 (#— a) 2 x— a 

and since these numerators are constant we may represent 
them by A', B', C', which gives 

A+Rx+C^f_ A' B' C 

(x—a) 3 ~~(x— a) 3 (x— a) 2 x — a 
which is the required form. 

As this demonstration may be applied to an expression 
containing any power of x, we make the proposition a gen- 
eral one, that 

< Vx m ' 1 —Qx m ^+ Rx+S 

(x—a) m ~~ 

A A' A" 

1 1 -4- qIc 

(x-a) m ' r (x-a) m ~ 1 ^(x-a)™-* 

Hence to integrate the expression 

P^ 4 +Q^ 3 +etc. 

(x — a) 3 (x — d) (x — c) 

we write 

/ A A! A' 1 D E \ 

\(x—a) 3 (x—a) 2 x—a x—dx—e-' 



PRINCIPLES OF INTEGRATION. 



273 



and reduce these fractions to a common denominator and 
find the values of A, A' , A\ D, E, in the manner already 
stated. We shall then have to find the integrals of the fol- 
lowing expressions, 

E D A" A' A 

aX * 7 iZX% (IX » / \ .-> (IX • / \ o (IX • 

x—e x—d ' x— a (x—a) 2 ' (x—a) 6 ' 
the three first we can integrate by the rule for logarithms 
and the others as follows. 

Since dx is the differential of x—a we will make x—a—zj 
then we have 

r Adx _ r Adz __ g _ _A_ _ A 

J (x—a) 3 J z 3 " 2s 2 2(3:— a) 3 

and 

/» A r dx _ rA'dz_ A/__ A' 
J (x—d) 2 ~~J z 2 z x—a 

Hence 

( p P.r 4 +Q^ 3 +etc . _ A A r ^ 

< J (x—a) 3 (x—d)(x—e) 2(x—a) 2 x—a 

( +A /7 log. (x—a) + Dlog. (x—d) + E\og. (x—e) 

(171) Let us take for example 

x 2 dx 
x 3 —ax 2 —a 2 x—a 3 
the denominator of this fraction may be resolved into the 
factors 

(x 2 —a 2 )(x—a)=(x—a)(x-Jra)(x—a) 
or 

(x — d) 2 (x-\-a) 
Making then 

x 2 A A! B 

(x— a) 2 (x+a)~~(x— a) 2 x— a x~\-a (1) 

and reducing the second member to a common denominator, 
we have 

x 2 _ A(x+a)+A r (x 2 -a 2 ) + B(x-a ) 2 

(x—a) 2 (x-\-a)~ (x—a) 2 (x-\-a) 



274 



INTEGRAL CALCULUS. 



Developing the numerator and making the coefficients of 
the same powers of x equal, we have 

A+B = i, A-2Btf=o, Aa- h! a 2 +¥>a 2 =o (2) 

from which we obtain 



hence 



and 



x 2 dx 



A,=$a, A'=|, B=J 



arfx $dx 



dx 



(x—a) 2 (x+a) 2(x—a) 2 4{x—a) 4(00 -\- a) 



/x a ax 
(x—a) 2 (x+a 



(*_,)>(*+«)- - 2 -^=7)+* l0 g- (*-«)+ilog. (x+a) 
( I 72) If all the factors of the denominator are equal the 
expression will be of the form 

x m ~ 1 dx 

{x — a) m 
and we may integrate this more directly. For let x—a^z, 
then dx=^dz and x=z-}-a, hence 

x m ~\ix _ (z+a) m -hfz 

(x—a) m ~ z m 

Expanding (z+a) 771-1 by the binomial theorem, we have 

■(z+a)™' 1 z m -^dz {m—\)az m -hh^ 

— j„ — — 1 



{111 — 1) {m — 2)a 2 z m ~^dz 
' 7z™ 



etc. 



[ r 



each of which terms may he integrated separately as in Art. 
(160) (162). 

Let us tak^ for example 

x 2 dx 
{x—d) 3 
Making x— a=z, then x=z-\-a and dx—dz, we shall have 

x 2 dx {z+a) 2 dz z 2 dz zazdz a 2 dz 
— ^ — ! — '- — 1 . .1 

(x—a) 3 z 3 z 3 z 3 z 3 

and 

{z+ a) 2 dz _ 2a a 2 __ 2a a 2 

? 3 "" S- ^ 2Z 2 ~ ^'^ X a ' x—a 2{x—a) 2 



i 



PRINCIPLES OF INTEGRATION. 275 

(173) When two differential functions are equal to each 
other it does not necessarily follow that their integrals are 
equal; but if they are not equal their difference will be a 
constant quantity for all values of the variable. In other 
words, if two functions have the same rate of change they 
will either be equal to each other constantly, or else their 
difference will be constantly the same. Thus the ages of 
any two persons increase at the same rate, and they will be 
therefore of the same age, or else the difference of their 
ages will always be the same. Two persons traveling in the 
same direction, at the same rate, will either be constantly 
together, or else there will be constantly the same distance 
between them. 

Hence in integrating the members of a differential equa- 
tion, it becomes an important part of the problem to ascer- 
tain if there be any difference between the integrals, and if 
so, what it is. 

To do this we add the indeterminate constant C to one 
member of the integral equation, which shall represent this 
difference if any. This is called the indefinite integral. 

Then, since the difference is constant for all values of the 
variable, we assign to the variable in one member of the equa- 
tion some value which will correspond to a known value of 
the other entire member, and thence obtain the value of C. 
That value having been substituted in place of C in the 
integral equation, will satisfy it for every value of the vari- 
able since it does for one value. 

To illustrate these principles, let us take the triangle ABC 
(Fig. 30). The differential of the surface of this triangle is 
axdx (Art. 62); a being the tangent of the angle CAB, and 
A the origin of coordinates. Hence the equation 

dS=axdx 
and integrating each member, and adding C to the second, 
we have 



276 



INTEGRAL CALCULUS. 



+ C 



(l) 



Now to determine the value of C we give to x a value cor- 
responding to a known value of S. But we know that at 
the origin in A where x=o, we have also S=o, and by sub- 
stituting these values in equation (1) we have 

o^o+C hence C=o 
and 

ax 2 

s= 



ax* 
2 



is the definite integral. 

If now we wish to know the value of any specific part of 
the triangle, such as ADD', we make #=.#'= AD, and we 
have 

_ax 2 _ xy _ AD XDD' 



22 2 

This is the specific integral. 

( I 74) We are not bound, however, to make the value of 
S commence at the origin where x=o. We may if we choose 
estimate it from any line as DD\ 
x=AD=x) we should have 



ax 



+C 



whence 



C=- 



ax 
2 



AD 



: — a~ 




and substituting this value inequation (1) we have 

— 2 
AD 



ax" 



2 2 

This is again the definite integral. For any portion of the 
triangle estimated from DD' we give the corresponding value 
of x, say ^=AE=„/, which gives 



PRINCIPLES OF INTEGRATION. 277 



Q-*V "2 '2\ AE ~ AD 

S — -(# 2 —x z )—a 



2 N 

for the value of the area DD'E'E. 

( I 75) There is another method of disposing of the inde- 
terminate constant, which consists in giving to the variable 
two definite values, and then subtracting one integral from 
the other. This is called integrating between limits. Thus 
in the case last noted, if we make x successively equal to 
x^-hX) and .#"=AE, we shall have 

ax * ax z 

S' = +C andS" = +C 

2 2 

and subtracting the first equation from the second we have 

2 V ' 2 

the constant C having disappeared in the subtraction. 

The notation for this kind of integration consists in plac- 
ing the two values of the variable at the extremities of the 
sign of integration ; thus 

fas" 
axdsv 

X 1 

indicates that the integral is to be taken between the two 
values of x represented by x" and x ; the subtractive one 
being at the lower extremity of the sign ; and the integral 
would be 

ax" % ax'* 



/x 



When the integral is to be taken for any particular value of 
x, as x', it would be written 



J x=x' c 



,axdx 
which indicates that the integral is to be taken where x—x\ 



278 INTEGRAL CALCULUS. 



EXAMPLES. 

Ex. i. Integrate 2xdx between the values of x=a and x=6. 

Ans. b 2 —a 2 
*b 



Ex.2. What is the integral of / $x 2 dx. Ans. b 3 —a z 
Ex. 3. Integrate / — x 2 dx. Ans. ~Ap z — # 3 ) 

Ex.4. Integrate / 2(e-\-x)dx. Ans. b 2 -\-2e{b — a) — a 2 
* a 

Ex. 5. Integrate / s(e-\-nx 2 ) 2 2nxdx. 

Ans. (e+nb 2 )^ — (e+na 2 ) 3 

/ b dx e + b 

— ; — . Ans, log. — ; — 

ae+x & e-\-a 

( I 76) INTEGRATION BY SERIES. 

If it be required to integrate a differential of the form 

F(x)dx in which F{x) can be developed into a series, the 

approximate integral may often be found by (Art. 164), and 

if the series is rapidly converging, its true value may be 

nearly reached. Let the differential be 

dx 
■ ; 9 —(l+X 2 )- 1 dx 

developing by the binomial theorem we have 
(i+^ 2 )" 1= i— x 2 +x± — ^ 6 +etc. 
and multiplying by dx and integrating we have 

/dx x z x 6 x 1, 

— ; — o=x— — + — ■— — + etc. 
I+x 2 3 5 7 

EXAMPLES. 

, dx 

Ex. 1 . What is the integral of — , — ? Ans. 

to i+x 



PRINCIPLES OF INTEGRATION. 279 



dx 

Ex. 2. What is the integral of r ? Ans. 

a — x 

dx 
Ex. 3. What is the integral of / _ ,, 2 ? Ans. 

dx 
Ex. 4. What is the integral of — , ? Ans. 

y 1—x 2 



(177) INTEGRATION OF DIFFERENTIALS OF CIRCULAR ARCS. 

We have seen (Art. 47) that if u designate the sine of an 
arc, then the differential of an arc will be 

die 



Vi-u 2 
hence the integral of the function of the form 

dx 



Vi-x 2 
will be an arc of which x is the sine. 
(178) If the expression is of the form 

dx 



Va 2 — x 2 
we may make x=av then 

x 2 —a 2 v 2 and a 2 — x 2 =a 2 — a 2 v 2=1 a 2 { < i— v 2 ) 
and 

dx=adv 
whence 



dx adv dv 



and 



VV— x 2 ^V i — v 2 Vi-z/ 
dx p dv 



/dx r 

V a 2 -x 2 ~~J Vi-v 2 

x . 
which is an arc of which z/=— is the sine. 

a 



280 INTEGRAL CALCULUS. 

( I 79) If x represent the cosine of an arc, then the dif- 
ferential of the arc (Art. 47) will be 

dx 



V i— x' 2 
hence the integral of the form 



dx 



V 1— x 2 

will be an arc of which x is the cosine. 

If the expression is of the form 

dx 



Va 2 —x 2 
it may be integrated as in (Art. 178) 

( I 30) If y represent the tangent of an arc then (Art. 47) 
the differential of the arc will be 

dy 

1-j-y 2 
hence the integral of a function of the form 

dx 
1-j-x 2 
will be an arc of which x is the tangent. 
(181) If the expression is of the form 

dx 
a 2 +x 2 
we may make x=av, whence 

dx=adv and a 2 +x 2 =a 2 +a 2 v 2 =# 2 (i + z> 2 ) 
whence 

P dx r adv _ 1 /» dv 

J a 2 + x 2= J a 2 (i+v 2 ) = ~aJ 1+v 2 

. 1 . . x 

which is equal to — into an arc of which v—~ is the tangent. 
1 a a to 

( I 82) If we represent the versed sine of an arc by z we 

have (Art. 47) for the differential of the arc 

dz 

V ' 2Z Z 2 



PRINCIPLES OF INTEGRATION. 281 

hence the integral of a function of the form 

dx 



V 2X — X 2 

will be an arc of which x is the versed sine. 
(183) If the expression be of the form 

dx 



V ' 2CLX — x 2 



we may assume x=av, whence 

dx—adv and 2ax— x 2 becomes 2a 2 v— a 2 v 2 
or 

a 2 (2v—v 2 ) 
whence 

dx c adv p dv 



/» dx __ J* adv _ I* 

J \/ On.T T 2 J &\/ 17) 7)2 J 



V2ax — x 2 J aV 2v—v 2 J V 



2V—V 



X . 

which is an arc of which v=— is the versed sine. 

a 



SECTION II. 



INTEGRA TION OF BINOMIAL DIFFERENTIALS. 

( I 84) The general expression for a binomial differential 

may be reduced to the form 

p_ 
x m ' 1 (a+bx n Yd (i) 

in which m and n are whole numbers, n is positive and x 
enters but one term of the binomial. 

For if m and n are fractional we may substitute another 
variable with an exponent equal to the least common mul- 
tiple of the denominators of the given exponents, which will 
then be reducible to whole numbers. 

If, for example, we have 

l_ v _p 

x 3 (a~\-bx^)^dx 
we make x=z Q , then dx—6z 6 dz, and we have by substitu- 
tion 

6zi(a+bz*)<idz 

If n is negative we can make x~— , and the expression 

would become 

j? J? 

x m - \a+bx- n ) v =z- m ~ 1 (a+bz n )<2dx 

in which the exponent of z within the parenthesis is pos- 
itive. 

282 



INTEGRATION OF BINOMIAL DIFFERENTIALS. 283 

If the expression is of the form 

x m -\ax r +l?x n )vdx 
we may divide the terms within the parenthesis by x r , and 

multiply the parenthesis by it, thus 

j> 
x m -\x T (a+bx n ' r )] vdx 
or 

pr p^ 

x m v(a+bx n - r )<idx 
thus we may secure the three stated conditions. 

P • 
(185) If ~ is a whole number and positive, the binom- 
ial may be expanded into a finite number of terms and 
integrated by Art. (163). If it is entire and negative the 
function becomes a rational fraction. 

examples! 

Ex. 1. Integrate the expression 

x 2 (a+bx*) 2 dx 
Expanding the binomial we have 

a 2 x 2 dx-\r2abx^dx-\-b 2 x 8 dx 
and integrating each term separately we obtain for the in- 
tegral of the binomial differential 

a 2 x 3 abx Q b 2 x* 

fx 2 (a-\-bx s ) 2 dx = + + 

j \ ^ ) 3 3 9 

Ex. 2. Integrate the expression 

x s (a-{-bx 2 ) s dx 

a^x^ -za 2 bx Q -zab 2 x 8 bx 10 
Am. — -+-^-7 +^— +— — 

40 o • IO 

Ex. 3. Integrate the expression 
x 4: (a-\-bx s ) 3 dx 

a s x 5 $a 2 bx 8 3<zb 2 x li b 3 x 14: 



Ans. 



11 14 



284 INTEGRAL CALCULUS. 

Ex. 4. Integrate the expression 

x 5 (a+b 2 x±) 3 dx 

a 3 x G 2a 2 b 2 x 10 lab+x 1 * b^x 1 * 

Ans. -T- + +~ + — 5- 

6 10 14 t8 

(186) Every binomial differential may be integrated when the 
exponent of the variable without the parenthesis, increased by one, 
is exactly divisible by the exponent of the variable within. 

To effect this we substitute for the binomial within the 
parenthesis, a new variable having an exponent equal to the 
denominator of the parenthesis ; thus in the expression 

x m ~\a+bx n Ydx (1) 

we make 

a-{-l>x n =zQ (2) 

then 

?L 
(a+bx n )<i=zP (3) 

From equation (2) we have 



x~ 



: v b ) 



and raising both members of the equation to the mih power 
we have 

m 

xm =\-jr) 

Differentiating and dividing by ??i we have 

2-1 
x mr-lfo=^[Ljj^ z^dz (4) 

and multiplying together equations (3) and (4) we have 

nb 



xn-^a + bx 71 )? dx= JL TzP +( i- 1 {— y- ) dr 



m . ... 

If now — is an entire positive number, this expression may 



INTEGRATION OF BINOMIAL DIFFERENTIALS. 285 

zv— a 

be integrated by raising — i — to a power consisting of a 

limited number of terms, and each term can be integrated 
separately. 

n 

If — be negative we may by formula D (Art. 214) increase 

the exponent until it become positive. 

EXAMPLES. 

(187) Integrate the expression 







3. 

x 3 (a+bx 2 ) 2 dx 




Assume 




a+bx 2 =z 2 




then 




(a+bx 2 )%=z* 

x* ^ ~ a 






b 




and 




zdz 

Vvll'i/i' 7 






Multiplyi 


n g ( x )> ( 2 )> (3)> together we 


have 






3. Z 2 

x s (a+bx) 2 dx =z^ 


— a 


of which the integral is 






z* 


az 5 (a+bx 2 ) 2 a 


{a+bx 2 ) 2 




?b 2 


$b 2 ~~ >jb 2 


5 b 2 


(188) 


Integrate the expression 








x 5 (a+bx 2 ) 2 dx 




Make 




a+bx 2 =z 2 




then 




(a-}-bx 2 ) 2 =z 





(1) 
w 

(3) 



(1) 



286 INTEGRAL CALCULUS. 



(2) 



b 

and 

zdz 
xdx=~y (3) 

Squaring (2) and multiplying by (1) and (3) we have 

1 ( z 2 —a\ 2 z 2 dz z Q dz—2az 4 -dz-\-a 2 z 2 dz 
x*{a+ bx 2 ) 2 dx=(—j—) -j~ = -y 6 

of which the integral is 



7^ 3 S^ 3 3^ 3 
and restoring the value of z we have 

1 {a+bx 2 )^ 2a(a+bx 2 ) 2 a 2 (a+bx 2 ) 2 
fx 5 (a+bx 2 ) 2 dx = ^3 ^ + ^3 

( 1 89) Integrate the expression 

x 5 (a-\-bx 2 ) s dx 
Make 

then 



(a+£* 8 )*=** (1) 



also 



b 
and 



(2) 



$z 2 dz 



2xdx— — ~ b — (3) 

Multiplying the square of (2) by (1) and (3) we have 
' x*(a+bx*)dx=^(^y^) z*d* 

=—j- s (z % — 2z s a + a 2 )dz 

30 1 °dz $z n adz ^a 2 z^dz 
= 2b 3 ~~~~b 3 + 2b 3 



INTEGRATION' OF BINOMIAL DIFFERENTIALS. 287 

which being integrated is 

3s 11 3<zs 8 $a 2 z 5 



22^ 3 Sb b iol? 3 
Substituting the value of z we have 



8 

~* 3 



2 ^{a+bx 2 ) ^- ^a(a+bx 2 ) s 3a 2 (a-\-bx 2 ) 
Jx*(a+bx*Ydx- -y s ^j + ^-3— 



( 1 90) Integrate the expression 

x s (a-\-x 2 ) 2 dx 
Make 

a+x 2 =z 2 
then 

x 2 =z 2 — a (1) 

(a+x 2 )~~ 2 =z~* ^ 

xdx=zdz (3) 

Multiplying together (i), (2), (3), we have 

x s dx(a+x 2 ) 2 = (z 2 —a)z~ 1 zdz= (z 2 —d)dz 
and integrating we have 

a 
z 3 (a+x 2 ) 2 1 

— —az — — a\a + x 2 ) 2 

3 3 V ' 

(191) Integrate the expression 

x 6 (a 2 +x 2 )~ 1 dx 
Make 

a 2 +x 2 =z 
then 

x 2 —z— a 2 (1) 

2xdx=dz (2) 

and 

(^+^ 2 )-i=^-i (3) 

Multiplying together (2), (3) and the square of (1) we have 

.t 5 ^+^ 2 ~Vj=- 

2 

which being expanded becomes 



INTEGRAL CALCULUS. 



z 2 — 2a 2 z-\-a 4: zdz 2a 2 dz cfidz 

>~ 1 dz= — — fc- 



2 2 2 20 

Integrating and substituting the value of s we have 

/^ 5 (^+^ 2 )-v^= ^^ +a ^ 2 -^ 2 (^ 2 +^ 2 )+ i 7iog.(^ 2 +^ 2 ) 

(192) Another condition under which a binomial differ- 
ential may be integrated is as follows : 

jp 
Put the expression x m ~ 1 (a+bx n )^dx into the following 
form 

^~ 1 [[~+b)x^dx 
or 

xm-^xz f^L+b)^dx=x * (ax- n +b)vdx 

By (Art. 186) this expression integrable when 

np 

m + — , 

q m p 

n n q 

is a whole number, hence 

A binomial may be integrated when the exponent of the vari- 
able without the parenthesis, increased by one, divided by the 
exponent of the variable within the parenthesis, and added to the 
exponent of the parenthesis is a whole number. 

EXAMPLES. 

(193) Integrate the expression 

a(i + x 2 ) 2 dx 
Make 

v 2 x 2 = i -\-x 2 
then 

(i+x 2 y%=z>-*x-* (i) 



INTEGRATION OF BINOMIAL DIFFERENTIALS. 289 

also 



V* — I 

whence 

dx ~x{v 2 -i) 2 v> 

and 

i=^4(^-i)3 (3) 

Multiplying together (i), (2), (3), we have 

3. awfr 

tf(i+.x 2 ) 2 dx — — -^ 

and by integration 

adv a_ ax 



V 1+x 2 
( I 94) Integrate the expression 

dx(a 2 +x 2 ) 2 =- 



Make 
then 

hence 



Va 2 +i 



v=x + V a 2+ x 2 



xdx x + \/ a z +x 2 

dv=ax+ , = — d x 

Va 2 -tx 2 Va 2 +x 2 



dv dx 



v Va 2 +x 2 
Representing the integral sought by X we have 

X ° = / ^^^=f-^-^og.v=\og.{x+Va 2 +.x 2 ) 

( I 95) Integrate the expression 

x 2 dx 



\U 2 +x 2 

Representing the integral by X 2 we have 

x 2 dx 

dX 9 = =■ 

2 Va 2 +x 2 



14 



290 INTEGRAL CALCULUS. 

Make 

v={a 2 x 2 -{-x 4 ') 2 
then 

a 2 xdx + 2x s dx a 2 dx 2x 2 dx 

dv— : — = — , +— = 

(a 2 x 2 +x± ) 2 V a * + x * Va 2 + x 2 

hence 

dv—a 2 dX -\~2dX 2 

where X has the same value as in (Art. 194). From this 

we have 

dv a 2 dX n 
dX 2 = — - 

* 2 2 

or 

x = g__ g 8 X 

2 2 2 

Replacing the value of X and v we have 

X 2 =~ 2 (a 2 + x 2 ) 2 - ^-log. (x + Va 2 +x 2 ) 
( 1 96) Integrate the expression 

x~ /L (i— x 2 ) 2 dx 
Make 

v 2 x 2 =i— X 2 
then 

#-3=2,2 + ! an d x-*=(v 2 +i) 2 (1) 

also 

x=(v 2 + i) 2 whence dx = — (v 2 + i) 2 vdv (2) 

and 

Multiplying together (1), (2), (3), we have 



INTEGRATION OF BINOMIAL DIFFERENTIALS. 291 

Integrating we have 

fx-Hi-x*) 2 dx=- — -v-- rVi-i 2 

*■ v ' 3 3.x 5 

(197) If a binomial cannot be integrated by any of these 

methods, there are others to which we may resort. These 

consist in making such a transformation of the expression 

that the exponent of the variable without the parenthesis 

or that of the parenthesis itself may be reduced so as to 

bring the differential into one of the integrable forms. This 

is done by separating the differential into parts, one of 

which shall be an integral quantity, and the other the form 

to which we desire to reduce the expression. This is called 

INTEGRATION BY PARTS. 

To effect this we resort to the principle on which the pro- 
duct of two variables is differentiated. We have seen (Art. 
11) that 

d{uv) = udv + vdu 
hence 

uv =J~udv -\-fvdv 
or 

fudv = uv — vdu ( 1 ) 

If now we can so transform the binomial differential, that 
while it is represented by the first member of the equation 
(1), it may also be represented in its transformed state by 
the second member, we see that the integral may be made 
to depend on that part represented by ydu y and that may be 
made to assume in certain cases the form of an integrable 
differential. 

The two general methods of doing this are, either to make 
the part represented by du to contain the variable without the 
parenthesis with an exponent diminished by that of the 
variable within the parenthesis ; or else to contain the par- 



292 



INTEGRAL CALCULUS. 



enthesis itself with an exponent diminished by one. In all 
other respects this part is to be identical with the given dif- 
ferential binomial. 

The following is the first of these methods. 
For convenience we represent the exponent of the par- 
enthesis by/, which is supposed to represent a fraction ; and 
substitute m for m— 1 ; and we have the general form 

d*{a+bx n )Pdx 
in which m and n are whole numbers. 
Make 

v=(a+bx n ) s 
in which s may have any required value. Differentiating 
we have 

dv=bnsx n ~\a-{-bx n ) s ~ x dx 

If we now assume 

x m (a-{-bx n )Pdx = udv 
we have 

x m (a+bx n )Pdx 



or 



~bnsx n ~\a + bx^^dx 

x m - n+ \a+bx n )P- s+1 
bns 



which being differentiated gives 
n — n J ri)x m -~ n { 
bm 
(p-s + i)x m (a+bx n )P 



(m—7t + i)x m - n (a +bx n )P- s+ * 

du = ** ' — 7 dx 

bns 



+ 



~dx 



but 



hence 



j (a+bx n )P- s+1 = (a+bx n )P- s (a+bx n ) 
I =a(a+-bx n )P- s +bx n (a-\-bx n y- s 



ra(m — n-\~i )x m ~ n (m + 1 + np — ns \x rl 1 , 
*=[ J J^~ +~ i '—\(a + ^)^ X 



INTEGRATION OF BINOMIAL DIFFERENTIALS. 293 

If now we take the value of s such that 
we have 



and 



m-\-i 



a(m — n^r i)x m ~ n (a+bx n )P- s 
du = J( . , ; — v^ dx 



Substituting these values of #, v, du and dv in equation (i) 
we have 

Formula A 

( fx m (a+6x.»)Pdx= ) 

\ x m - n+ Ha+6x n )P +1 — a(m — n + i)fx m - n (a+frx n )Pdx V 
( ~~ &(;ij>+m + 2) ) 

in which we find the integral of the given differential to 
depend on the integral of a similar differential in which the 
exponent of the variable without the parenthesis is dimin- 
ished by that of the variable within it. 

In like manner we should find 

fx m - n {a+bx n )Pdx 
to depend on 

fx m -* l (a+bx n )vdx 
and we may thus continue to diminish the exponent of the 
variable without the parenthesis as long as it is greater than 
that of the variable within it. 

( I 98) There is frequent occasion to integrate binomials 
of the form 

x m dx 



Va 2 -x 2 
Representing its integral by X m we have 

/» x m dx 



294 INTEGRAL CALCULUS. 

and substituting in the formula A (Art. 197) 

— 1 for b 

2 for n 

a 2 for a 

we have 

formula a. 



x m dx _(m— i)a 2 r x m ~^dx af 1 - 1 ^ 
Va 2 -x 2 ~ m J Va 2 -x 2 ~ 
(199) Integrate the expression 



x m dx _{m— i)a 2 r x m ~*dx rt™- 1 / — 



adx 
dX (] 



\/a 2 — x 2 

We have found (Art. 47) that the differential of the arc of 
a circle is equal to 

R^/sin. 



VR 2 -sin. 3 
hence we have 
X =arc of a circle of which a is radius and x is the sine. 
(200) Integrate the expression 

/x' 2 dx 

Substitute in formula a, 2 for m and we have 

dx 



a~ p ax x , 

X.= — / —===—Va 2 -x 2 

2 2 J \/ a 2_ x 2 2 



which is equal to 



— X — Va 2 -x* 

2 u 2 



where X has the same value as in (Art. 199). 

Similarly by substituting different values for m in formula 
a we have 



INTEGRATION OF BINOMIAL DIFFERENTIALS. 295 



C x*dx 2a 2 x 3 / 

X 4 = / — t===-Xo - — Va 2 -x 2 
4 J \ a 2 -x 2 4 a 4 

r x*dx za 2 x 5 ,— - 

J \a 2 —x 2 4 6 
.T s ^r 7<2 2 # 7 



/• x~ax iu •»- /—z 

x s = — == = '— x„~— V^ 2 - 

47 V a 2 — # 2 < 



r ,* 3 - 

^"— X" 

in which the values of X , X 3 , X 4 , X 6 , X 8 remain the same 
throughout. Thus formula a reduces the integral of a dif- 
ferential of the form 

x m dx 



Va 2 —x 2 

to that of one depending on the integrals of differentials of 

the forms. 

x m -\ix x m ~\ix x m ~ Q dx 

and so on 



V a 2 — x 2 ' Va 2 — x 2 ' Va 2 — x 2 
until, if m is an even number, we shall after™ operations 
find the integral of the given differential to depend on that 
of a differential of the form 

dx 



y a 2 - x 2 

x 
which is the differential of the arc of a circle of which — 

a 

is the sine (Art. 177). 

(201) By a similar substitution in formula A we may find 



formula b 

thus 

x m dx _«* m ~ 1 ./-^ — 7; {m—i)a 2 /» x m ~\ix 
Xra 



/• x" b ax x"° x /— Km—i)a~ /» 

= f , — = Va 2 + x 2 - - Z — /- 

J V^ 2 +x 2 m nt J < 



Va 2 + x 2 

If then we have the expression 

/x^dx 



Va 2 + x 2 



296 INTEGRAL CALCULUS. 

we would make m=4 in formula b which would then become 
; 3 . 'xa 2 c x 2 dx 



X 



x 6 , xa~ /» 



4 4 J Va 2 + 



x 



2 



The integral of 

x 2 dx 



Va 2 +x 2 
we have found (Art. 195) to be 



1 a* 



— {a 2 + x 2 ) 2 — — log. (x+Va 2 +x 2 ) 

hence 

/» x^dx _x s 3a 2 x 3# 4 

/ - y ■ — V^ 2 +* 3 ~~ — — y^s + ^2+— log.(^ + y a* +*s) 

(202) The expression 

x m =j - 



X m dx 
V 2ax— x 2 



may be integrated by first reducing it to the given form (Art. 
184) and making the proper substitutions in formula A (Art. 

197)- 

It may however be integrated by an independent process 

as follows : 

Make 

v=x™~W 2ax __ x 2 ^ax^-^-x^y 

and we have by differentiating 

#( 2 7/z — 1 )x <2m ~'*dx — mx jlm ~ 1 dx 
dv = 7 ■ 

(2 ax*™- 1 — x* m ) 2 

which becomes by dividing the terms by x 771 - 1 

a{^2m — 1 ) x m ~ ^dx mx m dx 
dv= — — j 

(2(ZX—X 2 ) 2 {2CIX —x 2 ) 2 

Now this last term is equal to 

m . dX m 

hence 

a{2 m — 1 )x m ~ 1 dx 
dv= 7 — — m .dX m 

(2ax—x 2 ) 2 



INTEGRATION OF BINOMIAL DIFFERENTIALS. 297 

or by transposition 

a(2m—i )x m ~ L dx dv 
dX.m—~~ \ — ~~~ 

m\ax—x*y 

and by integrating and substituting the value of v we have 

formula c 
x m ~ dx a(2m—i) /» x m ~ 1 dx 



/x m ~~ dx _a\2m—i) /» x m ~ 1 dx x m ~ L , 
^2a — x 2 ~ m J V2CIX—X 2 m 
an expression which depends on the integral of 

x m ~ 1 dx 



2ax— x 2 



X 2CLX— X 2 



in whicn the exponent of the variable without the parenthe- 
sis is diminished by one. 

(203) If we take the expression 

adx 



</X — 



V 2ax — x 2 

we see (Art. 47) that it is the differential of an arc whose 

radius is a and whose versed sine is x ; or, which is the same 

x 
thing, an arc whose versed sine is — and radius 1 ; hence 



■.=!-. 



adx 



V 2CIX— X* 

(204) If we take the expression 

xdx 



ver. sin. 



-1. 



x 



x,=/- 



° V 2dX— X 2 

and make m in formula c equal to 1, we shall have 

Xi = X — V 2CZX— X 2 
in which X has the same value as in (Art. 203). 
Similarly 

P x 2 dx $a x , 

X 3 — / — 7 == — X T — — V 2ax — x' 

J V2CIX-X 2 2 * 2 

and 



298 INTEGRAL CALCULUS. 

/x z dx za x 2 , 

—^=-===— Xo - — V 2ax- x 2 
Vzax-x 2 3 8 3 

where X t , X 2 , X 3 , have the same value throughout. 

Thus formula c reduces the binomial differential. 

x m dx 



V 2CIX — X 2 

to depend successively on the integrals of 

x m ~ 1 dx x m ~ <2 dx x m ~^dx 



V 2<2X— X 2 ' V 2(ZX— X 2 ' V 2aX— X 2 

and finally on 

dx 



V 2CIX— X 2 

which represents the differential of an arc whose versed sine 



x 
is — as we have seen. 
a 



(205) To find the integral of 



3. 

x 2 dx 



V 2dX — X 2 

we substitute in formula c, f for m which gives 
3 _i .1 



/x'dx 4a /» x~ax 2X~ , 
' f = = / , =^ — V 2CIX— X 2 
\/ ?.rr..r, — .t. 2 3 " V vox. — x 2 3 



x 2 dx 4a /» x 2 dx 2X 2 

\ /, 2ax— x 2 3 ^ V 2ax— x 2 3 

Dividing the terms of the first fraction in the second mem- 

.1 
ber of the equation by x 2 we have 



1 

x 2 dx dx 



V 2CIX— X 2 V 2CI — X 

of which the integral is 

~~ 2 V 2<Z — X 

hence 



/ 



3. 

x 2 dx 8a , 2X , 

-V 2a — x — — V 2a— x 



V 2ax— x 2 3 3 



INTEGRATION OF BINOMIAL DIFFERENTIALS. 299 

(206) The method of diminishing the exponent of the 
variable without the parenthesis by means of formula A, 
will of course only apply when m is positive. But we may 
obtain from this another formula which will diminish the 
exponent when it is negative. To do this we multiply the 
formula A by the denominator and we have 

{ b (;ip + 7/i + i)fx m (a+6x n )P= ) 

\ x m - n+1 (a + bx n )P +1 ~a(77i—7i + i)fx m - n (a+bx n )Pdx \ 
or 

r fx m ~ n (a+bx n )P dx — \ 

\ x m - n+i (a+bx n )P i ' 1 — b{7iJ> + 772 + i)fx m (a + bx n )Pdx I 

{ a(m—n-\-i) ) 
Making m — n=—m we have 

Formula B 

I/x~ m {a + bx n )P dx = \ 

x- m+ ^a+bx n )P +1 — b(7ip + m+7i + i)fx- m+n (a+bx n )dx > 
a( — 771+1) ) 

If tn denote the greatest multiple of n contained in 771 we 
shall have after /+i reductions the integral of 

x~ m (a+bx n )Pdx 
to depend on that of 

x -m+ a+ i)n( a +fix n )Pdx 

and if — ?n+(f+i)n=n— -1 we shall have (Art. 165) 

r *, x (a+bx n )P +1 

fx^Ka+bx-Ydx ^^^ 

but in this case 

— 771+1 

—t 

n 

a whole number ; and hence the original expression may be 

integrated as in (Art. 186). 

(207) To find the integral of 



300 INTEGRAL CALCULUS. 



tx 1 



x 2 (i+x 3 )' 6 
Substitute in formula B 

2 for m 
i for a 
i for b 

3 for n 

and we have 

since a{ — ;/z + i) = — i and b(np— m + i) = i. 
(208) To find the integral of 

—x~ 2 (2— x 2 )~%dx 



si 

X 2 (2—X 2 ) 2 

Substitute in formula B 

2 for m 
2 for a 
— i for b 
2 for n 

-I for/ 

which gives 

fx~ 2 {2— X 2 ) 2 dX——X~ 1 (2—X 2 ) 2 — 2f(2 — X 2 ) 2 dx 

since #( — z0 + i) = — 2 and b{ivp—?ti-\-n-\-\) — 2. 

(209) Besides the method of reducing the exponent of 
the variable without the parenthesis, we may make the 
integral to depend on that of another expression of the same 
form in which the exponent of the parenthesis itself is re- 
duced by one. This is the second general method referred 
to in (Art. 197). 

Let us make v=x s where s is an exponent to which we 
may assign any required value. From this we obtain 

dv=sx s ~ 1 dx (1) 



INTEGRATION OF BINOMIAL DIFFERENTIALS. 301 

If now we assume 

udv=x m (a+bx n )P dx (2) 

we shall have by dividing equation (2) by equation (i) 

u — {a-\-bx)P 

and 

7/i—s + i , . bnp 



du 



x m - s (a+Z>x n )Pdx+—x m - s+1 (a+frx n )P- 1 x n - 1 dx 



but 

(a+bx n )P =(a+bx n )(a+bx n )P- 1 

hence 

a(m—s + i)+b(m—s-\-i+np)x n , N „ - 

afc= -^ ; r; — x™-Ha+bx n y-idx 

Let the value of s be taken such that 
m—s-\-\ -\-np"=>o 
or 

and we shall have 



np+m+i 

Substituting these values of #, v, du> dv in formula (i), (Art. 
197), we have 

Formula C 

„ , x x m+1 (a+bx n )P +anp/x m (a+bx n )P~ 1 dx 
/x m (a+6x n )Pdx= ; *. , 

in which the integral of the expression is made to depend 
on that of one of the same form in which the exponent of 
the parenthesis is one less than that given. 

By a similar process this last may be made to depend on 
the integral of one whose exponent of the parenthesis is 
again one less ; and so on until the exponent of the paren- 
thesis shall have become less than one. 



302 INTEGRAL CALCULUS 

(210) To integrate the expression 

d*Va 2 +x 2 
substitute in formula C 

o for m 
a 2 for a 
i for b 
2 for n 
\ioxp 
and we obtain 






but by (Art. 194) we have found 

dx 

Va 2 + . 
hence 



/dx 
-===log. (x+V^+^) 



X\/ a 2 _i_ x 2 a 2 

<dxVa 2 + x 2 = — +— log. {x + \/ a 2 + x 2\ 

2 2 7 

in like manner we find 



X\/ X 2_ a 2 a 2 

fdx^x 2 -a 2 = — log- {x+Vx 2 - a 2 ) 

22 ' 

(211) If the exponent of the parenthesis is negative, this 

formula will, of course, not answer, but we can easily deduce 

from it one that will effect the object. For this purpose we 

clear it from fractions, transfer the integral term, and divide 

by the coefficient of the last term in formula C and we have 

Formula D 

C Jx m (a+bx n )P~ 1 dx ) 

\ _ -x m +\a+bx n )P +(np + m + i)Jx m {a+bx n )P dx V 
\ anp j 

(2 1 2) To find the integral of 

(2— x 2 ) 2 dx 



INTEGRATION OF BINOMIAL DIFFERENTIALS. 303 

substitute in formula D 

o for m 

2 for a 
— 1 for b 

2 for n 
-1 for p-x* 



and we have 



f{2-x 2 ) %dx=-( 2 -x 2 ) *= : 



2 " J 2V2-X 2 

(2 1 3) To find the integral of 
xdx 



-j—x(i-{-x 3 ) z dx 



(1+* 3 ) 3 
we substitute in the formula 

1 for m 
1 for a 
1 for b 
3 for n 
— \ for/— i 
and obtain 

1 "V 2 2 

y^:(i+^ 3 ) s dx = — — -(i+x 3 ) 3 + 2jx(i+x 2 ) 3 dx 

2. 

in which ^(i+jc 3 ) 3 ^ may be developed into a series, and 

each term integrated separately. 



SECTION III. 



(2 1 4) Application of the Integral Calculus to the Measure- 
ment of Geometrical Magnitudes . 

We have seen (Art. 173) that when two differentials are 
equal, their integrals will also be equal or else have a con- 
stant difference. It is upon this principle that the method 
of measuring geometrical magnitudes by means of the cal- 
culus is founded. We obtain the expression for the rate of 
change in the magnitude, in a function of one variable and 
its differential. It will follow that the magnitude itself is 
equal to the integral of the function, or else the difference 
between them will be constant for all values of the variable. 

Thus let M represent any magnitude, and let F(x)dx rep- 
resent its rate of increase while being generated by its ele- 
ment — that is, its differential : F(x) being the differential 
coefficient, and a function of x. 

Then we have 

dM=F(x)dx 
which is an equation between two differentials, hence the 
integrals are equal or else differ by a constant quantity. If 
we represent the integral of T(x)dx by X we shall have 

M=X + C 
where C represents the constant difference between the 
quantities whose rates of change are equal. 

The method of disposing of the term C is shown in (Art. 
173), and the result will be an expression for the value of 



MEASUREMENT OF GEOMETRICAL MAGNITUDES. 305 

M in terms of one variable. Then assigning to this variable 
any specific value, we obtain the value of M from the be- 
ginning up to that value of the variable; or, by giving to 
the variable two successive values, the difference of the two 
resulting expressions will give the value of that portion of 
M lying between the two values of the variable. 

RECTIFICATION OF CURVES. 

(215) To rectify a curve is to find what would be its 
length if it were developed into a straight line ; in other 
words, to find the measure of its length. When its differ- 
ential can be obtained in an integrable form it is said to be 
rectifiable. 

The general expression for the differential of any plane 
curve whose equation is referred to rectangular axes is 

(Art. 34) 

^ = Vdx 2 +ay 2 
and hence 

v = fVdx 2 +dy 2 '+C 

is the general expression for an indefinite portion of any 
such curve. In order to obtain the integral of this expres- 
sion, we must know the relation between x and y which we 
obtain from the equation of the curve; and by means of it 
eliminate one of the variables and its differential from the 
formula; thus producing a differential function involving 
but one variable and its differential, whose integral, when it 
can be obtained, will be the length of an indefinite portion 
of the curve. 

(216) To find the length of a Circular Arc. 

We have in (Art. 47) several expressions for the differential 
of an arc of a circle, in terms of its trigonometrical func- 
tions, which already contain but one variable in each case. 



306 INTEGRAL CALCULUS. 

If we select that in which the tangent is the variable, we 

will represent it by / and the formula becomes 

dt i 

du =' — r^r = ,, 2 dt 

Developing the fraction we have 
i 



i-/ 3 +/ 4 -/ 6 +/ 8 -etc. 



i+t 2 ~ 
hence 

du— — r-rdl=dt—t 2 dt+t±dt—l Q dt+etc. 

and integrating each term separately we have 

/ 3 t* P / 9 
fdu—u=t— — + — — — — — + etc. 
J 3 5 7 9 

or 

t 2 / 4 / 6 t s 
u=l(i — — + — — — + — — etc.)+C 
v 3 5 7 9 ; 

But we have found (Art. 180) C=o, and if we assume 

^=30° 
we shall have 

and substituting this value for t we have 

/— . 1 1 1 1 

U =V i( J ~ +Z " — 2 — „ 3 +— *T— etc. 

8V 3 • 3 5 • 3 7 • 3 3 9 • 3 4 

which being reduced is equal to 0.523598 nearly, for the 
length of an arc of 30 ; and multiplying by 6 we have the 
arc of a semi-circle equal to 3. 141 5 88 when radius is 1. 
Hence this is the ratio between the diameter and the entire 
circumference. 

(217) To jiitd the length of an Arc of a Parabola. 

We have found (Art. 57) that the differential of an arc of 
a parabola is 

<ty r-i — 7 

du—^—yp^+y* 



MEASUREMENT OF GEOMETRICAL MAGNITUDES. 307 

The integral of this (Art. 210) is 

U =2-VJ1T7+ , log. (y+Vf*+y*)+C 

If we estimate from the vertex of the curve where u=o and 
v=o we shall have 



hence 



o=£\og.p+C 



C=-|log./ 



Substituting this value of C we have for the definite integral 

»=^-v^+7+|iog. (j+V/^+7-|io g ./ 

or 



(218) 7# 7?/^ //*<? length of an Arc of an Ellipse. 

We have found (Art. 57) that the differential of an arc of 
an ellipse is 

A V A 8 -* 2 

If we take c to represent the distance from the center to the 
focus of the ellipse we have 

B»=A»-«" 
hence 



, 1 , /A* - c*x 3 

If now we represent the eccentricity of the ellipse by e we 
have c=Ae, and hence 



. 1 , /A 4 -AVjc 2 
du——r-ax\ 



/A*—AVj 



308 INTEGRAL CALCULUS. 

or, dividing by A 4 under the radical and multiplying by A 2 
without it we have 



du—Adx\ - 



A 2 — x 

€" X" \ " 

Developing (i — ~vr) by the binomial theorem we have 



i_ 



1 

„2 ^-2 2" 



e'~x 2 \ 2 e 2 x 2 e^x^ y^x 6 

~i — — ;nr— - — a ; — — — 7 — r^— etc. 



/ e~x~\~ 



2A 2 2 . 4 . A 4 2 . 4 . 6 . A 

hence 

Adx e 2 x 2 dx <? 4 x^dx 

du — 



VA 2 -x 2 2 &' VA 2 -x 2 2.4A 3 • Va 2 -x 2 
3<? 6 x6dx 



2 . 4. 6A 5 " <\/A*-—x* 
Making 



-etc. 



Adx x 2 dx x^dx 



vl^^-vF^-^'Tx^-^' etc - 

we have 

/^=X -^X 2 - 7 -^ X¥ X 4 - 2 - - 6A5 X 6 -etc. (1) 

Now by (Art. 199) X = the arc of a circle of which A is 
the radius and x the sine, and (Art. 200) 

A X j 

X 2 =-X --Va 3 -jc 3 
3 2 ° 2 

also 

3 A 



-x 



2 



x 4 =- — x 2 -— Va 2 - 

4 4 2 4 

and 

zA 2 x 5 , 

X f = J *g-X 4 -.-gVA«-*» 

If we make x=o and estimate from the extremity of the 
conjugate axis, we have x=o and C=o. If we make x~=>A 
we shall have w=- a quadrant, and since 



MEASUREMENT OF GEOMETRICAL MAGNITUDES. 309 

we have 

A 3A 2 3A 3 3 . ^A 5 

o — A ft , A, — .A. 9 — A n , A fi — ^^-» 

8 2 0> 4 ^ 4 2.4 0> 6 2.4.6® 

and substituting these values in equation (i) we have 

u=XJi— — — 7 — > — etc.) 

uv 2.2 2.2.4.4 2.2.4.4.6.6 7 

for one-fourth of the circumference of an ellipse; X being 

one-fourth of the circumference of a circle of which the 

diameter is equal to the major axis of the ellipse. 

Hence the whole circumference is equal to 

2~A(i — — — ■ — 2 — 7— etc. 

v 2.2 2.2.4.4 2.2.4.4.6.6 

It will be seen that as the eccentricity diminishes the cir- 
cumference of the ellipse approaches the value of 27rA, 
which it reaches when e=o, and the curve becomes a circle. 

(219) To find the length of the Arc of a Cycloid. 

We have found (Art. 129) that the differential equation of 

a cycloid is 

ydy 
dx— , ==- 

V 2ry — y 2 

By substituting this value of dx in the formula we have 



du~- 



■ Vdx* +dy* =dy\/ y% 9 + 1 =dy\/ 
V 2rv — v V 



2ry 



2ry — y* v 2ry — v 3 



or 

u=^~2rfdy{2r—y) ^ = — 2\/2r{2r~yY —■— 2 ^ 2r{2r' —y+C 
If we estimate the arc from D (Fig. 55) where y=2r we 
shall have u—o and C=o, and hence making j/=FG 

^ = D'F = — 2V 2;- (2r—y) (1) 

We see from the figure that 

D'E'= 2 r and D'H.' ==2r—y 
hence 



3IO INTEGRAL CALCULUS. 



V2r(2r-j)=VD r E' . D'H'=D'F' 
so that the arc of a cycloid is equal to twice the corres- 
ponding chord of the generating circle. 

If we take the arc DA, the corresponding chord of the 
generating circle becomes the diameter D'E', and half the 
arc of the cycloid is equal to twice the diameter of the gen- 
erating circle, or the entire arc is equal to four times that 
diameter. Thus, if we make y~o we have 

z/— 4/' 
or 

D'FA = 2 D'E' and ADB= 4 D'E' 

(220) To find the length of the Arc of a Logarithmic Spiral. 

We have found (Art. 77) that the differential of an arc of 

a polar curve is 

du = \/r 2 dv 2 +dr 2 

and the equation of the logarithmic spiral is 

z'~Log. r 

Hence 

Mdr M. 2 dr 2 

dv= and dv 2 '- 



r r* 

Substituting this value of dv 2 we have 



du = VMdr 2 + dr 2 =drV M 2 +1 
In the Naperian system M = i and 

du—dr\/ 2 
hence 

u=r\/lz + C 
If we estimate the arc from the pole where r=o we shall 
have C=o and 

u—r\/ 2 
That is, the length of an arc of a Naperian logarithmic 
spiral estimated from the pole is equal to the diagonal of a 
square of which the radius vector is the side. 



MEASUREMENT OF GEOMETRICAL MAGNITUDES. 311 

(221) To find the length of an Arc of the Spii'al of 'Archimedes. 

The equation of the spiral (Art. 84) is 

r—av 

in which a= — ■ and z> = the arc of the measuring circle whose 

radius is the value of r after one revolution. Hence 

du=V ' r 2 dv 2 +dr 2 =advV 1 + v 2 
the integral of which may be found in (Art. 210). Substi- 
tuting 1 for a and v for x, thus 

u=a{ V ** V +i!og. ( V +VI+J*))+C 

Estimating the arc from the pole where #=0 we shall have 
C^o and 

u=-^[v+Vi+v* +log.(v+ V I+7 2 )] 

(222) To find the length of an Arc of a Hyperbolic Spiral. 

The equation of this spiral is (Art. 86) 

ab 
rv=ab or r—~ 



Differentiating we 


have 






V 










dr=-- 


abdv 

V 2 






whence 














du- 


■V* 


Pdv* 


a 2 b 
~+ V 2 


2 
-dv 2 — 


abdv t x 
" v V^ 


+ 1 


and 




u — 


abfv~ 2 


a^VV 








+ 1 




Integrating by formula 


B (Art. 


206) 


making in 


the formula 








VI- 


-2 












a- 


-1 












b = 


-1 












/= 


-1 

■ 2 












n= 


-2 







312 INTEGRAL CALCULUS, 

we have 



fv~ 2 dvV v 2 + i=-v- 1 (i+v 2 ) 2 + 2fdv{i+z> 2 ) 
or (Art. 221) 

= _ g -i( I+P »)f +2 [ £V I + 7 ' s +xiog, (r+ VT+^)] 

hence 



* =<rf[— sr-^i +^ 2 ) 3 +vV i + v 2 -\- log. (z>+V 1 + v 2 )\+C 
Estimating the arc from the point where v==-o we have 

a 

u=a&(jy) 2 = vo 

which is as it should be, since from the equation of the 
curve, when v=o the radius vector is infinite. As v is infi- 
nite when r=o we shall have u=o at the same time. 
Hence the curve is unlimited in but one direction. We may, 
however, find the length of any intermediate portion by sub- 
stituting the two corresponding values of v in the integral 
function and taking the difference of the results. 

(223) Quadrature of Curves. 

The quadrature of a curve is the process of finding the 
measure of a plain surface bounded wholly, or in part, by a 
curve. 

To find the area of such a surface we must find its dif- 
ferential in a function of one variable, which being integrated 
will give an expression for an indefinite portion of the area, 
from which any specific portion may be obtained by assign- 
ing corresponding values to the variable. 

(224) To find the area of a Semi- Parabola. 

We have (Art. 65) for the differential of the surface of a 

parabola ± 

d S =ydx = y/ 2px 2 dx 

of which the integral is 



MEASUREMENT OF GEOMETRICAL MAGNITUDES. 313 

3 

But when x=o we have S=o, and hence C=o; so that 
the surface of a parabola bounded by the curve, the axis 
and an ordinate is equal to two-thirds of the rectangle 
described on the ordinate and corresponding abscissa. 

(225) To find the area of any Parabola. 

The general equation of the parabola is 

y n =ax 

from which we obtain 

ny n ~ 1 dy 

dx = 

a 

hence 

b-Jydx-J a -( n+1 ) a - n+1 *y+^ 

If we estimate the curve from the origin where S=o we 
have x=o, and hence C=o, and 

b— — : — XV 
n + i y 

That is, the area of that portion of any parabola, bounded 
by the curve, the axis and the ordinate, is equal to the rec- 
tangle described upon the ordinate and corresponding 

. . . n 

abscissa, multiplied by the ratio , . If /z~2, as in the 

common parabola, we have 

S=%xy 
If «=f, as in the cubic parabola, we have 

If ji=i, the figure becomes a triangle and we have 

S=±xy 

or half the base into the height. 

(226) To find the area of a Circle. 
We have (Art, 63) for the circle 

ydx =dx\l R2 x 2 



15 



314- INTEGRAL CALCULUS. 

Making R = i we have 

dS=ydx=dxV i— x 2 =dx{i—x 2 ) 2 

Developing the binomial and multiplying each term by dx 

we have 

x 2 dx x^dx x Q dx $x 8 dx 

dS=dx— — — ^~ — 7 — rr - — etc. 

2 8 io 128 

from which we obtain by integrating each term separately 

iA/ \A/ %A/ S fV 

S=#— —7-— — — — — etc.+C 

6 40 112 1152 

Estimating the area from the center where x=o we have 

S=o, and, therefore, C=o, so that the series expresses the 

area of any segment between the ordinate at the center 

where x=o and the ordinate corresponding to any other 

value of x. Hence if we make x — i we have the area of a 

quadrant equal to 

T 1 1 1 5 pfr. 

1 6 TO 112 1152 eic - 

which by taking enough terms may be reduced to 

•78539 
Hence the entire area of the circle will be equal to 

3-!4i5 6 
equal to ~ where radius is 1. 

(227) We may also find the area of a circle by consider- 
ing it as being described by the revolution of the radius 
about the center. In this case the radius of the circle be- 
comes the radius vector and we have (Art. 82) 

r 2 dv 

dS= — 5- 
2R 

where v represents the arc of the measuring circle and R its 

radius. Integrating the terms of this equation we have, 

since r is constant, 

Estimating the area from the beginning where S=o we have 
z'=o, and hence C=o, and 



MEASUREMENT OF GEOMETRICAL MAGNITUDES. 315 



s =%w (0 



" 2 V 

"2R 

is the measure of a sector of a circle of which z/=the meas- 
uring arc. Making K=r we shall have Z7=the arc of the 
given circle, and equation (i) becomes 

rv 
2 
that is, the measure of a sector of a circle is half the pro- 
duct of the radius into the arc of the sector ; and hence for 
the entire circle, the area is equal to half the product of the 
radius into the circumference. 

If we make z>= the entire circumference we have 

z; = 2-R 
and substituting this value in equation (i) we have 



2R 
that is the area of a circle is equal to the square of the 
radius multiplied by the ratio between the diameter and the 
circumference. 

(228) To find the area of an Ellipse. 

We have in the case of an ellipse (Art. 64) 

B 7 

d S =-ydx = -rdxv A 2 — x 2 

hence 

S=^f(A*-x 2 )idx 

Integrating by formula B (Art. 209), and substituting 

o for m 

A 2 for a 

— 1 for b 

2 for n 

\ for/ 

we hare 



316 INTEGRAL CALCULUS. 



1_ 



but (Art. 178) 

Va 2 -x 2 a 



~r n x-i p dx . x 

j(A 2 -x 2 ) 2 dx= /— ==zn sm -i- 



hence 



B , AB x 

S=- T xVA*-x 2 +-~~sin-i-T+C 

2A 2 A 



Estimating from the center where x=o we have S=o, and 

hence C^o. Making then x=-A we have 

AB . AB - 

S= — sm." 1 !— — . — 

2 22 

for one-fourth of the area of the ellipse, since the arc whose 
sine is 1 is equal to one-fourth of the whole circumference ; 
and we have for the area of the entire ellipse 

S=~AB 
We may also observe that (Art. 63) 
dxV'A^x 2 
is the differential of the area of a circle whose radius is A, 

B 

hence the area of an ellipse is tX the area of the circum- 
scribing circle which is -A 2 ; and is, therefore, equal to 

B 

-£. -A 2 =-AB 

If A^B the expression becomes 

-A 3 or -R 2 
for the area of a circle. 

(229) To find tJte area of a Segment of a Hyperbola. 

We have in the case of a hyperbola 

B 7 - 

y-^Vx 2 -A 2 

whence 



MEASUREMENT OF GEOMETRICAL MAGNITUDES. 317 



d$=ydx=-rdxV x*— A 2 



A 

Integrating by formula C (Art. 209) we have 
B , . AB 



S=-—Xy / x 2_ A 2_t_ 1q < X + V S X 2__ A 2\ +C 
2A 2 ^ 

To find the value of C we make x=A where S=o, and we 
have 



hence 



AB 

:-— log. A + C 



AB 
C= — log. A 

2 ° 



and 

2 A 2 to V \ / 

which represents a portion of the area between the curve 
and the ordinate lying on one side of the axis. Hence the 
area of the entire segment cut off by the double ordinate is 
B 



W^=Ai,AB log.(*+ V * 2 - Aa ) 



but 



B 7 — 

y x~ — A 2 —y 



A 

hence 

S=*y-AB log. (|+|)=*y-AE log. {^ Z ~ f ) 

for the value of the area. 

(230) We may also find the area of that part of the sur- 
face lying between the curve and the asymptotes, by using 
the equation of the hyperbola referred to its center and 
asymptotes, which is 

xy =z m 
but as the asymptotes are not usually at right angles to each 
other, we must introduce into the expression for the differ- 
ential of this area, the sine of the angle which they make 



318 INTEGRAL CALCULUS. 

with each other (Art. 58) which we will call v. We shall 
then have 

mdx 
^/S^sm. v.ydx=sin. v 

and 

S^sin. v .m. log. x+C 
If we call the abscissa of the vertex 1, and estimate from 
the corresponding ordinate, we shall have at that point 

m=i, S=o, log. x=o and hence C^o 
And since sin. v may be considered as the modulus of a sys- 
tem of logarithms, we may make 

S = M . log. .T=Log. x 
That is, the area between the curve and the asymptote, 
estimated from the ordinate of the vertex, is equal to the 
logarithm of the abscissa, taken in a system whose modulus 
is the sine of the angle made by the asymptotes with each 
other. 

(231) To find the area of a Cycloid. 

We have (Art. 129) 

ydy 

dx= , = 

V 2ry—y* 

hence 

y 2 dy 
dS =ydx = . = 

V 2ry— y 2 

Integrating this by formula c (Art. 202) we have 



Z r +y A 



S=|r . ver. sin. -1 j— — —\ / 2ry— y 2 +C 

Estimating the integral from A (Fig. 76) where jy=o, we 

have 

S=o and hence C=o 

and taking the integral where 

y=2r=DE 
we have 

Ttr 2 
S = fr . ver. sm.-" 1 2r=3 




MEASUREMENT OF GEOMETRICAL MAGNITUDES. 319 

that is, the area ADE is equal to three times the semi-circle 

DF'E. Hence the entire area of the Q Q P 

cycloid is equal to three times the area 
of the generating circle. 

(232) Another method of obtaining 
the area of a cycloid is, to consider that 
portion of the rectangle ACDE which Fig. 7 6. 

lies outside the curve. 

If we make'GF = 2r— jp=z> we shall have the differential 
f the area DCAF equal to vdx or 

d^ =(2r—y)dx=(2r—y)—y-- =dyV 2ry—y 2 

V 2ry—y* 

Now if we take the equation of a circle with the origin at 

the extremity of the diameter we shall have 

yCtOC — uX'y 27" X — X 

which is the differential of the segment of a circle of which 
x is the abscissa. Hence dy\ y 2 ry— y 2 ls trie differential of 
a circle of which y is the abscissa, that is of the segment 
F'BE. Hence the two areas ACGF and F'BE have the 
same rate of change or differential for the same value of y ; 
and since they are both equal to zero when y=o, they are 
equal for every other value of y (Art. 173), and, of course, 
when y = 2r. Hence 

ACD=DF'E=— 

2 

But the rectangle ACDE =717- . 2r=27zr 2 1 hence 
ADE^ACDE-ACD- 3 -^ 



as we found in (Art. 231). 

(233) To find the area bounded by the coordinate axes and 
the logarithmic curve. 

We have had (Art. 137) for the logarithmic curve 

Mdy 



#=Log. y and dx=- 



y 



3 2 ° 



INTEGRAL CALCULUS. 



hence 

dS=ydx=Mdy and S=My+C 
If we estimate the area from AD (Fig. 56) where ;=iwe 
have 

o=M+C 
whence 

C=-M 
and 

S=M(jf~i) 
If we make y=o we have 

S=z-M=area ADD' 
If y — 2 =RO we have 

3 = M=area ADRO 
So that although the axis of abscissas is an asymptote (Art. 
88) to the curve on the negative side, and, therefore, will not 
meet it within a finite distance, yet the area enclosed be- 
tween them is limited and equal to ADRO. 

(234) If we take the curve represented by the equation 



y 



2 = J 

X 



to which the axes of coordinates are asymptotes, we shall 
find a case somewhat similar. 

Putting the equation into the form ^ 
1 



x = - 



r 



we have 



and 



hence 



dx = - 



2dy 



2dy 
d S =ydx=—™2 



K 


\ ° 






H 




sF 


E 






K 






B 



Fig. 



s=-+c 

y 



MEASUREMENT OF GEOMETRICAL MAGNITUDES. 32.T 

If we estimate the area from the line AC where 7=00 we 
have 

o=o + C 
hence 

2 
C^o and S=— 

y 

If we makej/ = i=FT we have 

S= 2 =ATDC 

that is, the area ATDC is equal to twice the square AHFT, 
and is, therefore, finite, although the curve FD does not 
meet the axis AC at a finite distance. 

If we take the area between the limits y = i and y—o, we 
shall have 

S=f-2 

that is, the area FEBT is infinite, although AB is likewise an 
asymptote to the curve. 

(235) To find the area described by the radius vector of the 
Spiral of Archimedes. 

The differential of the area of a polar curve (Art. 82) is 

r 2 dv 

v being the arc of the measuring circle and R its radius. 
The equation of the Spiral of Archimedes (Art. 84) is 

r—av 
hence 

dr—adv 
or 

dv = — 
a 

Hence, making R = i we have 

/r 2 dv fr 2 dr r % 
=/ =7T-+C 
2 J 2a 6a 

If we make r=o we have 
21 



$22 INTEGRAL CALCULUS. 

S=o and hence C=o 

r 

and since a=— we have 
v 

r 2 v 

Making v—2- we have for the area described by one revo- 
lution of the radius vector 

S=— 
3 
that is, the area described by one revolution of the radius 
vector is one-third of the area of the circle described with a 
radius equal to the last value of the radius vector. 

If the radius vector make two revolutions we have v—qn 
and 



where r =2r and 



3 
But in making two revolutions, the radius vector describes 
the first part of the area twice. This, therefore, must be 
subtracted, and we have the area enclosed by the curve and 
radius vector after two revolutions equal to 

3 ' 3 r 3 r 

and by subtracting the first again we have the increased area 
described during the second revolution equal to 

3" r 3 r — z r 

After m revolutions we have 



where r f "=-mr^ hence 

m~m?r* nfi~r 2 , % 

s=^^=-— - (i) 

3 3 

Subtracting from this the area described by the radius vec- 
tor during (m—i) revolutions we have 



MEASUREMENT OF GEOMETRICAL MAGNITUDES. $2$ 

S'=^0»8-( OT -i)3) ( 2 ) 

Substituting {m+i) in place of m we have 

S*=y((/*+i)B_,*3) (3 ) 

for the area described by the radius vector during the 
(m+i)th revolution. Taking the difference between equa- 
tions (2) and (3) we have the additional area described by 
the {m+i)th revolution of the radius vector, that is the area 

lying between the nith and \jn-\-\)th spires thus 

2 

S ;/ -S'=— ((w + i) 3 -2;// 3 +(w-i) 3 )^2;^r 2 

We have found the additional area described by the radius 
vector during the second revolution equal to 2-r 2 , h^nce the 
additional area described during the (jn-\-i)th revolution is 
equal to m times that described by the radius vector during 
the second revolution. That is, the increase of the additional 
areas described by the radius vector during successive revo- 
lutions, is itniform and equal to twice the area of the circle 
described with a radius equal to the radius vector after one 
revolution. 

If the area ABP (Fig. 34) be required, that is, the addi- 
tional area corresponding to the arc BC described after the 
first revolution, we shall have 

27T 
V = 2~+ ■ 

n 

and 

r 

r'=r+~ 
n 

and the required area will be 



-}"- 



3 v n» 3 



or 



324 INTEGRAL CALCULUS. 

_ -(n + 1) r 2 (n+i) 2 ~r 2 
3// *. n 2 "3 



(;z + i) 3 - 



3^ 3 o 

Developing (/z + 1) 3 we have 

or 

-r 2 , 3 3 1 \ ~r 2 ~r 2 , 1 1 \ 

s= — (i+-+-4r+-r)- — = — (1+-+ — r) 

If BCD = J circumference = — , then 73=4 and 
ABP=-J-«(i+i+A) 

(236) To find the area of the surface described by the radius 
vector of the Hyperbolic Spiral. 

The equation of the hyperbolic spiral (Art. 86) is 

ab 
rv—ab or r= — 

in which # is the radius of the measuring circle and b is the 

unit of the measuring arc. Hence 

pr 2 dv r ab 2 dv ab 2 

S=/ — =/— — = - — ■ 

J 2a J 2V M 2V 

which is infinite when z>=o, and zero when ^=00. If we 
make v=b—AB (Fig. 38), and v=^b=As, we shall have 

ab ab PB . OR 

S=afr— — — — = 

22 2 

(237) To find the area described by the radius vector of a 
Logarithmic Spiral. 

We have (Art. 87) for this spiral 
^=Log. r 



MEASUREMENT OF GEOMETRICAL MAGNITUDES. 325 

and 

Mdr 

dv :=1 

r 

Substituting this value in the formula, and making R = i and 

M^i, we have 

fr 2 dv P rdr r 2 

S= / -p-= / — =— +c 

J 2K J 2 4 

If we estimate from the pole where S^o we have r=o, and 
hence C— o, and 

S=— 
4 

That is, the area described by the radius vector of the Nape- 
aian logarithmic spiral is equal to one-fourth of the square 
described upon the last value of the radius vector. 

(238) Areas of Surfaces of Revolution. 

We have seen (Art. 66) that the differential of a surface 
of revolution, where the axis of. revolution is the axis of 
abscissas, is 

S:= f 2 ~yVdx 2 +dy 2 
the radical part being the differential of an arc of the re- 
volving curve. 

To apply this formula to a particular case we must obtain 
from the equation of the revolving curve, the value of one 
variable and its differential in terms of the other, so that, 
when substituted in the formula we may have the differen- 
tial of the surface in terms of one variable which can then 
be integrated. 

(239) To find the convex surface of a Cone. 

We have (Art. 67) for the differential of the convex sur- 
face of a cone 

d S == 2 ~axdx V a 2 -j-i 

in which x is the length of the axis and a is the tangent of 



326 INTEGRAL CALCULUS. 

the angle made by the revolving element of the cone with 
the axis of revolution. Integrating we have 

S=-0#VaM-i+C 
Estimating from the vertex where S=o we have x=o, and 
hence C^o and 

S=~ax 2 \/ a 2 + I 

But from the equation of the generating line we have 

y=ax 
and hence 



and by substitution we have 



X 

or (Fig. 35) making x=KB 



SrzzrrCD^ABS+CD 3 
that is, the convex surface of a cone is equal to the circum- 
ference of the base multiplied by half the slant height. 

(240) For the convex surface of a cylinder we have 

j=R=radius of the base 
hence 

S =f2-yVdx* +dy' z =/27:Rdx= 2'Rx 
that is, the convex surface of a cylinder is equal to the cir- 
cumference of its base into its altitude 

(24 1 ) In the case of the sphere we have (Art. 68) 

dS = 2~'Rdx 
hence 

Estimating from the center where x=o we have S=o and 
hence C=o, and the measure of an indefinite portion of the 
convex surface of a sphere is 

S = 2ttR^ 
the same as that of the circumscribing cylinder having the 
same altitude. 



MEASUREMENT OF GEOMETRICAL MAGNITUDES. 327 

Making ^=R we have 

S = 2 -R 2 

for the measure of the convex surface of half the sphere ; 

hence for the entire sphere we have 

S= 4 -R3 

or four great circles. 

(242) To find the surface of an Ellipsoid described by an 
ellipse revolving about its major axis. 

Making 



Vdx 2 +dy 2 =du 
we have 

2nyV dx 2 +dy 2 —2-xydu 
But we have found (Art. 218) 

Adx r e 2 x 2 e^x^ y % x^ 

du ^l^\2- x A 1 ~2~K 2 ~~ 2 . 4 A±"~~2 .'4 . 6A 6 ~~ etC ') 
hence 

2nAvdx e 2 x 2 e^x 4 - $e Q x Q 

' /S ~ VA*^*^ 1 ""*^*" * • 4 • A 4_ 2. 4 .6.A 6 ~" etC ^ 
But 

Ar_ 
VA 2 —x 2 ' 
hence 

e 2 x 2 e^x^ $e Q x Q 

dS = 2~~Bdx(j.—~ 7~2~ ^T-~ — " — 2 — Tr — etc.) 

v 2 . A 4 2 . 4 . A 4 2.4.6.A 5 J 

Integrating each term separately we have 

e 2 x 2 e^x* 3<? 6 ^ 6 

S = 2^B^(i- 7 -y 7S j- 2>4S#A4 - 2i4#6>7>A6 -etc+C 

Taking the integral between the limits 

x=o and x=A 

we have for half the surface of the ellipsoid 

e 2 e^ te® 

S = 2ttBA(i- — — - — ^—, -etc.) 

v 2.3 2.45 2.4.6.7 J 



=B 



328 INTEGRAL CALCULUS. 

and multiplying this expression by 2 we have the measure 
of the entire surface of the ellipsoid. 

If we make A=B, then e=o 9 and we have for the surface 
of the sphere 

S= 4 7rR 2 
as before. 

(243) To find the surface of a Paraboloid of revolution. 
We have (Art. 69) in the case of the paraboloid 

Integrating according to (Art. 165) we have 

s=-p(j 3 +/ 2 ) f +c 

Estimating from the vertex where S=o andj=o we have 



hence 



and 



°=-^ 3 + C=y/ s +C 



_2iep* 



s=Yp {{f+p2)% ~ p%) 



(244) To find the area of the surface described by a Cycloid 
revolving about its base. 

We have (Art. 129) in the case of the cycloid 

V 2ry—y 9 
hence 



vsm^=V^-.+*'=*v / ^t. 



MEASUREMENT OF GEOMETRICAL MAGNITUDES. 329 



and by substitution 



dS = 2~yV dx 2 +dy 2 '-=-2-ydy\/ ■ 



2ry 

2ry—y 2 



or 

y 2 dy 



i~V 2r I ~ 



V 2ry—y 2 
But we have found (Art. 205) 



/: 



y 2 dy 8r , 2y 



V 2r — y--~ — V 2T— y 



V ' 2ry-y 2 3 3 

hence 

8r,— *y 



S = 2-V 2r)— — V 2r— y— — V 2r—y)+C 

If we estimate the surface from the plane passing through 
the middle point of the base we shall have S=o when y=-2r, 
hence C^o. Then making y := o we have for half the sur- 
face required 

, — f Sr ,— ■ 



— V 2r)=2£-r 2 



3 



and for the entire surface twice that quantity. That is, the 
area of the surface described by the revolution of a cycloid 
about its base is equal to twenty-one and one-third times 
that of the generating circle. 

(245) The Cubalure of Solids. 

The cubature of a solid is to find the dimensions of an equiv- 
alent cube or other known volume. 
We have (Art. 71) 

7iy 2 dx 
for the differential of a solid of revolution where y is the 
ordinate and x the abscissa of the bounding line of the 
revolving surface which generates the solid ; and the axis of 
abscissas is the axis of revolution. Hence 

V=f~y 2 dx 



33° INTEGRAL CALCULUS* 

To apply the formula to any particular solid or volume, we 
eliminate one of the variables by means of the equation of 
the bounding curve, thus producing a differential function 
of one variable which may be integrated. 

(246) To find the volume of a Right Cone. 

Making the vertex the origin we have 

y^ax 
for the equation of the bounding line ; but a is the tangent 
of the angle made by this line with the axis of the cone, and 

b . 

is equal to y, where b is the radius of the base and h the 

the length of the axis ; hence 

b b 2 

y=-j* and y 2 =jpx* 

whence 

b 2 b 2 x^ 

Estimating from the origin we have V=o, #=o, and hence 

C=o, and making x~-= h we have for the entire cone 

h 

V=~b 2 - 

3 

that is, the volume of a cone is equal to one-third of the 
product of its base by its altitude, or equal to one-third of 
a cylinder of the same base and altitude. 

(247) To find the volume of a Sphere. 
From the equation of the circle we have 

hence 

V=f-y 2 dx=f7:(RZ--x 2 )dx=-(R2x-~)+C 
Estimating from the plane passed through the center, where 



MEASUREMENT OF GEOMETRICAL MAGNITUDES. 33 1 

x=o, we have S=o and C=o, and making x=K we have 
for half the volume of the sphere 

V=f~R 3 
and for the entire sphere 

Since the surface of the sphere is equal to 4ttR 2 we have 
the volume equal to the surface multiplied by one-third of 
the radius. 

(248) To find the volume of an Ellipsoid. 

Taking the origin at the extremity of the transverse axis 
we have for the equation of the bounding line or curve 

B 2 
y 2 =*-^{2Ax— x 2 ) 

hence 

/B 2 B 2 x* \ 

-^( 2 A*-**)^=^(A*s-— )+C 

Estimating from the origin where x=o we have V=o, and 
hence C=o ; and making x=2 A we have 

V=^( 4 A3-|A3)=r:iB 2 A-^B 2 . 2A 

that is, the volume of a prolate ellipsoid is equal to two- 
thirds of the volume of a cylinder having the minor axis for 
its diameter, and whose altitude is equal to the major axis. 

If the ellipse is made to revolve about its minor axis we 
should have 

V'=7:fA 2 B 
for the volume of an oblate ellipsoid, and hence 

V: V'::tt|B 2 A:^A 2 B::B:A 
that is, the volume of a prolate ellipsoid is to that of an 
oblate ellipsoid generated by the same ellipse, as the minor 
axis is to the major axis. 



332 INTEGRAL CALCULUS. 

If A=B we have 
as before. 



V=f*R3 



(249) To find the volume of a Paraboloid. 

In + his case we have 

y 2 =2px 
and 

N —f-y 2 dx~2~pfxdx — 2~p =7zJ>x 2 

Estimating from the vertex where x=o we have V=o, and 
hence C=o, and 

r 2 

or, the volume of a paraboloid is equal to half the volume 
of the circumscribing cylinder. 

(250) To find the volume of a Solid described by the revolu- 
tion of a cycloid about its base. 

Since in the case of the cycloid 

ydy 



dx~ 



V 2ry—y 2 
we have 



V 2ry — y 2 
But we have found (Art. 204) 

y z dy _ $r y 3 

2ry — y 

x 2 =— x t - z V 
^ 2 x 2 

^i~^o V 2ry —y 



J V 2ry—y 2 3 3 



3 r v y A/ 

2ry—y z 



X. = arc of a circle of which r is radius and y the versed 
sine. 



MEASUREMENT OF GEOMETRICAL MAGNITUDES. 333 

Integrating between the limits y—o and y=^2r we shall 
have half the volume required; but ^=o gives V=o and 
C=o and^ = 2r gives 

X =*r 



X 1 =X =«r 

y 

X 2~1T X 1 



V «. Z~r* 



and 



S^ v y 3 w », s^ 2 ^ 



hence the entire volume is 

But the volume of the circumscribing cylinder is 

4-r 3 . 2-r=8^ 3 r 3 
Hence the volume of the solid generated by the revolution 
of a cycloid about its base is five-eighths of that of the cir- 
cumscribing cylinder. 



Appendix. 



APPENDIX. 



Geometrical Fluxions. 

It has been said that the " reductio ad absurdum " or method 
of exhaustion of the ancient mathematicians contains the 
germ of the differential calculus. This is an error. There 
is nothing in that method that has any affinity to the true 
principle of the calculus. The method of rates, in the sim- 
ple and obvious meaning of the term, is as remote as possi- 
ble from the method of exhaustion. Its demonstrations are 
direct, logical and conclusive. No absurd hypothesis are 
admissible- and therefore no " reductio ad absurdum." There 
is neither exhaustion nor limits, nor any idea to which these 
methods have any sort of affiliation. 

Moreover the true principles of the calculus are so sim- 
ple and so easily applied that if they had occurred to these 
men they could at once have seized and used them without 
the aid of Al-gebra, and thus have avoided the "tedious and 
operose reductio ad absurdum " altogether. The principles of 
this science have been so exclusively associated with the 
forms of analysis, that it has come to be considered as purely 
analytical in its character. This is indicated by the term 
"calculus" itself, as well as by the terms "transcendental 
analysis " and " calculus of functions. " But the truth is these 
15 337 



3$8 APPENDIX. 

principles are wholly independent of analysis, and may be 
applied as directly to the geometry of Euclid and Archi- 
medes as to that of Descartes. Those propositions that 
require the " tedious reductio ad absurdum" or the absurd 
method of the infinitely sided polygon, may be easily and 
directly solved by them without resorting to the abstractions 
of analysis. 

These principles are contained in the method of rates, 
which is in fact their development. As applied to geometry 
they are two, viz. : 

First. The rate of increase of any geometrical magnitude, 
while being generated, may be me astir ed by a suppositive increment 
arising front the uniform movement of the generatrix, at the 
existing rate, during a unit of time, in the direction to which it 
may be then tending; and, therefore, such suppositive increase 
may be taken as a symbol to represent that rate. 

Second. If two magnitudes begin to be at the same moment, 
and the ratio of their rates of increase is constant, the ratio of 
the magnitudes themselves will be constantly the same as that of 
their rates. 

Thus if two persons set out at the same moment and 
place to travel in the same direction, at constant rates, the 
ratio of the distances traveled by each will constantly be the 
same as that of their rates of travel. If one travel twice as 
fast as the other he will always be twice as far from the 
starting point. 

Now to apply these principles to the measurement of geo- 
metrical magnitudes. 

Proposition I. 

(252) To find the area of a Circle. 

We will suppose the circle to be generated by the revolu- 




APPENDIX. 339 

tion of radius CA about the center C at a 
uniform rate. When the radius is in the 
position CA, and revolving toward B, every 
point in it will tend to move in a direction 
perpendicular to CA, and hence the point 
A will tend \.o describe the line AB tangent ^FlgT^ 

to the circle and perpendicular to CA ; and if left to its ten- 
dency would describe that line at a uniform rate. The^line, 
therefore, may be taken as the symbol representing the rate 
of increase or generation of the circumference of the circle. 
But while the point A tends to move in the direction AB, 
every point in the radius CA tends to move in a direction 
parallel to it, and at rates proportional to their distances 
from the center C. Hence the radius itself, if left to its ten- 
dency when at CA, would be found at CB, when the point A 
is at B ; and the triangle CAB would be generated at a uni- 
form rate during the same time that the line AB is generated. 
The triangle may, therefore, be taken as the symbol of the 
rate at which the area of the circle is generated, and the 
ratio of these symbols is also the ratio of the rates which 
they represent. But the triangle is equal to |CA . AB, that 
is the ratio between the rate of generation of the circumfer- 
ence and that of the area of the circle is half radius ; and 
this being constant is the ratio between any part of the cir- 
cumference and the corresponding part of the circle through- 
out their generation, and, of course, when it is completed. 
Hence the area of the circle is equal to half the radius into 
the circumference. 

Proposition II. 

(253) To find the area of the convex sitrface of a Cone.- 

Suppose the cone to be generated by the revolution of the 
triangle ADC (Fig. 79) about the axis DC. The hypothe- 



340 



APPENDIX. 




Fig. 79. 



neese AD will generate the convex 
surface, and the point A will gener- 
ate the circumference of the base. 
When the triangle is in the position 
ADC and revolving towards E, the 
point A if left to its tendency at that 
instant would describe the line AE, 
perpendicular to AC, in some unit of 
time, and hence AE may be taken to 
represent the rate at which the circumference of the base is 
generated. Now every point in the line AD tends to move 
in a direction parallel to AE, and at a rate proportional to its 
distance from the axis DC ; hence if left to that tendency 
the line AD would describe the triangle ADE, and be found 
at DE in the same unit of time. Hence ADE (Art. 251) 
may be taken to represent the corresponding rate of 
generation of the convex surface of the cone. But 

ADE AD 
ADE=AE . |AD or — rrr = , that is, the ratio between 

the rates of generation of the convex surface of the cone 
and the circumference of its base is constant and equal to 
half its slant height. Hence the ratio between the magni- 
tudes generated will be the same (Art. 251), and their con- 
vex surface divided by the circumference of the base equals 
half the slant height, or, the convex surface equals the cir- 
cumference of the base multiplied by half the slant height. 

Proposition III. 

(254) To find the measure of the volume of a Cone. 

The cone being supposed to be generated by the revolu- 
tion of a right angled triangle about one of its sides, which 
becomes the axis of the cone, while the base is generated 
oy the other side as its radius ; let us suppose the generating 
triangle to have arrived at the position ADC (Fig. 79), the 



APPENDIX. 34I 

point A moving towards C. Then every point in the trian- 
gle tends to move in a direction perpendicular to its plane 
and at a rate proportional to its distance from the axis CD ; 
so that if AE is taken to represent the line that would be 
described by the point A in a unit of time in consequence 
of that tendency y than at the end of the same unit of time 
the line AD would be found at ED, and the triangle ADC, 
at EDC, so that the pyramid DAEC would be the volume 
generated by the triangle, during the same unit of time and 
may therefore (Art. 251) be taken to represent the rate at 
which the cone is generated ; while at the same time the 
triangle ACE would be described by the radius AC of the 
base and would, therefore, represent the rate at which the 
base of the cone was generated. But the volume of the 
pyramid DAEC is equal to its base ACE multiplied by one- 
third of its altitude DC. Hence the rate of generation of 
the cone divided by that of its base = a constant quantity. 
Therefore (Art. 251) the cone itself divided by its base is 
equal to the same quantity being one-third of its height — or 
the volume of the cone is equal to its base multiplied by 
one-third of its height. 

Proposition IV. 

(255) To find the area of the surface of a Sphere. 

Suppose the sphere to be generated by the revolution of 
the semicircle CBD (Fig. 80) about the diameter CD. Then 
the semi-circumference CBD will 
generate the surface of the sphere. 
Now every point in the curve CBD 
tends to move in a direction perpen- 
dicular to its plane at a rate propor- 
tional to its distance from CD, the 
axis of revolution, and und»er this 
tendency it would in some unit of time 




342 APPENDIX. 

assume the position of the semi-ellipse CED, generating at 
the same time the convex surface of the ungula CEDB, 
which is, therefore, the symbol (Art. 251) representing the 
rate of generation of the surface of the sphere, while the line 
EB described by the point B, in the middle of the CBD, and 
perpendicular to its plane is the symbol representing the cor- 
responding rate of generation of the circumference of a great 
circle. But the convex surface of ungula is equal to its ex- 
treme height EB, multiplied by the diameter CD, which is, 
therefore, the constant ratio between the rates of generation 
of the surface of the sphere and of the circumference of its 
great circle. The magnitudes themselves are, therefore 
(Art. 251), in the same ratio, and the surface of a sphere is 
equal to its diameter multiplied by the circumference of its 
great circle. 

Proposition V. 

(256) To find the measure of the volume of a Sphere. 

The sphere being supposed to be generated by the revo- 
lution of the semicircle CBD about the diameter CD (Fig. 
80), when it is revolving towards the point E, every point in 
it will tend to move in a direction perpendicular to its plane, 
and at a rate proportional to its distance from CD the axis 
of revolution; and the point B in the middle of the arc 
CBD will tend to describe the line BE perpendicular to the 
plane of CBD, in some unit of time^ and would do so if left 
to that tendency. The semicircle CBD, at the end of the 
same unit of time, would be found in the ellipse CED, 
having described or generated the ungula ECBD, which 
may, therefore (Art. 251), be taken as the symbol of the rate 
at which the volume of the sphere is generated (Art. 251), 
while EB is the symbol of the rate at which the circumfer- 
ence of its great circle is generated. But the volume of the 



APPENDIX. 343 

ungula is equal to its extreme height EB multiplied by two- 
thirds of the square of the radius, which is, therefore, the 
ratio between the rates of generation of the volume of the 
sphere and of the circumference of its great circle. Hence 
the magnitudes themselves are in the same ratio (Art. 251), 
and the volume of the sphere is equal to the circumference 
of its great circle multiplied by two-thirds of the square of 
its radius — or the circumference, multiplied by the diameter 
(equal to the surface), multiplied by one-third of the radius. 















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